A bijective toolkit for signed partitions
William J. Keith
Abstract. The recently formalized idea of signed partitions is examined with intent
to expand the standard repertoire of mappings and statistics used in bijective proofs
for ordinary partition identities. A new family of partitions is added to Schur’s Theorem and observations are made concerning the behavior of signed partitions of 0 in
arithmetic progression.
Mathematics Subject Classification (2000). Primary 05A17; Secondary 11P83.
Keywords. partitions, signed partitions, partitions of zero, complement, Ferrers diagram.
1. Introduction
In a recent article [2], Andrews examined the notion of partitions in which we allow some
parts to be negative, calling these signed partitions and producing extensions of some
classical theorems of the subject. His proofs were all analytic: short generating function
manipulations that work for negative exponents. This paper begins exploring the toolkit of
bijections and combinatorial statistics for its use with negative parts. We start by reproving
Andrews’ theorems bijectively, usually with refinements, and then look further abroad.
For example, we can apply bijective techniques to add an intermediary class of signed
partitions to a theorem of Schur, giving a short bijective proof of several of its classical
clauses:
Theorem 1. Partitions of n into distinct nonmultiples of 3 are equinumerous with signed
partitions of n in which all positive parts are divisible by 3, each at least 3 times the
number of positive parts, with negative parts not divisible by 3, all less than 3 times
the number of positive parts and having distinct 3-weights, but −(3(ki + 1) ± 1) both
appearing only if 3(ki + 1) does.
Formally, we can regard a signed partition of an integer n as a finite sequence
(λj , . . . , λ1 , λ−1 , . . . , λ−k ) where λi ∈ Z \ {0}, λj ≥ · · · ≥ λ1 > 0 > λ−1 ≥ · · · ≥
λ−k , and λj + · · · + λ−k = n. (There are many equivalent ways of viewing such partitions, each with their own context and utility. We shall encounter several in this article.)
2
William J. Keith
Considered, as Andrews did, in their context from the work of Euler, these new partitions
are quite old objects. However, perhaps due to analytic concerns for the convergence of
the generating function – there is, after all, an infinite set of such partitions for any n, if
we do not restrict the kinds of parts allowed – little research has been done on them in the
intervening centuries.
This may have been a loss to the subject. As Andrews points out, several classical
partition identities can be cast in this light. Perhaps more usefully, once our eyes have been
opened signed partitions by other names appear in various places in the literature. Garrett
[7] discusses Laurent-polynomial generalizations of the Rogers-Ramanujan identities and
others from Slater’s list, using lattice paths that start below the x-axis in a manner that
cries out for combinatorial interpretation in terms of signed partitions. In a representationtheoretic context, they can be interpreted as the signatures of analytic representations of
GL(n) (as in [12]). Study may thus be more fruitful than previously imagined.
In Section 2 we reprove bijectively a related group of four theorems from [2], refining some of them in doing so, and from there expand our search to other identities. In
Section 3 we take note of the techniques we are using and define them for signed partitions in general. We demonstrate some utility by adding a morsel to a theorem of Schur
to produce a short bijective proof. Possible adaptations of the Ferrers diagram for partitions are discussed, along with uses of the diagram such as conjugation, profiles, and
complementation. We close with some open questions.
The standard notation in q-basic hypergeometric series is used throughout:
(a)n = (a; q)n = (1 − a)(1 − aq)(1 − aq 2 ) . . . (1 − aq n−1 )
(1)
(q)n = (q; q)n
(2)
(a)∞ = (a; q)∞ = lim (a; q)n
(3)
n→∞
2. Four Theorems on Odd and Distinct Parts
The four theorems of interest from [2], appearing there as Theorems 7 through 10, are as
follows:
Theorem 2. Let E(m, n) denote the number of ordinary partitions of n with exactly m
even parts, none of which is repeated. Let S(m, n) denote the number of signed partitions of n in which each positive part is larger than the number of positive parts, the
negative parts are all distinct, each not exceeding the number of positive parts, and m
designates the excess of the number of positive parts over the number of negative parts.
Then E(m, n) = S(m, n).
Theorem 3. Let G1 (n) denote the number of signed partitions of n in which each positive
part is even and at least twice the number of positive parts, and the negative parts are
distinct and odd with each smaller than twice the number of positive parts. Let G2 (n)
A bijective toolkit for signed partitions
3
denote the number of ordinary partitions of n in which each part is ≡ 1, 4 or 7 (mod 8).
Then G1 (n) = G2 (n).
Theorem 4. Let S1 (n) denote the number of signed partitions of n with an odd number
(say j) of positive parts, each positive part ≥ (j − 1)/2, with distinct negative parts each
odd and < j. Let S2 (n) denote the number of ordinary partitions of n into parts not
divisible by 4. Then S1 (n) = S2 (n).
Theorem 5. Let R1 (n) denote the number of signed partitions of n with an odd number
(say j) of positive parts, each positive part ≥ (j − 1)/2 with distinct negative parts each
≤ (j − 1)/2. Let R2 (n) denote the number of ordinary partitions of n into parts that are
either odd or ≡ ±2 (mod 10). Then R1 (n) = R2 (n).
There are some clear commonalities among all these theorems. Each deals with
partitions into odd parts, and partitions into distinct parts, two sets of partitions with several commonly known bijections relating them. In each set of signed partitions we are
equipped with a number of positive parts each exceeding a given minimum, and negative
parts distinct and restricted in size and therefore number. These sizes are small enough
that, after at most minor manipulation, the negative part of the partition is dominated by
the guaranteed portion of the positive part. Visually, it “fits within” the guaranteed positive part, and immediately the procedure suggested to us is to subtract the negative parts
from the guaranteed positive parts, leaving a pair of ordinary partitions. To construct a
bijective proof of the theorems, it remains then merely to combine the pair into a single
partition in a way that records the identities of the original pair.
Proof of Theorem 2. This theorem is provided with an extra statistic our bijection must
preserve – that the number of even parts in the ordinary partition is the same as the excess
of the number of positive parts over negative parts in the associated signed partition. This
statistic will appear again in our refinements of the other theorems, and the additional
structure makes it a good theorem to display the underlying intuition in detail.
Begin with an element of the set of signed partitions counted by S(m, n). Suppose
that the number of positive parts is j. In an extension of the usual Ferrers diagram, in
which we view the parts of an ordinary partition as rows of lattice points in the fourth
quadrant of length equal to each part, we can view a signed partition as rows of lattice
points first and third quadrants, thus:
In Figure 1 we have a diagram of an element of S(3, 70) with 8 positive and 5
negative parts, the signed partition λ = (17, 15, 12, 11, 10, 10, 9, 9, −1, −3, −4, −7, −8).
We have highlighted the portion of each positive part equal to the number j of positive
parts (which in the future, following Andrews, we will call the + number, and similarly k
the − number). We are guaranteed that this square and the column following it are fully
occupied. There is also highlighted a square of equivalent size which by the definition of
S(m, n) must contain the negative parts. Indeed, since the negative parts must be distinct,
if bottom-justified in the square they would fit below its diagonal.
We would like to transform this signed partition into an ordinary partition with 3
distinct even parts, and it seems reasonable that we should do so in some fashion clearly
starting from the number of positive parts, and reducing by the number of negative parts.
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William J. Keith
F IGURE 1. An element of S(m, n).
It is suggestive at this point to observe the final step in the generating-function proof of
the theorem, which employs the identity
qj
2
+j j
z (1 +
1
1
1
)(1 + 2 ) . . . (1 + j ) = (−zq)j q j(j+1)/2 .
zq
zq
zq
The combinatorial interpretation is clear: we take the j 2 + j required minimum
rectangle of the positive partition, and split it along its diagonal into two triangles of base
j. One we leave in the positive portion, where as a standard construction the addition of
j to the largest part, j − 1 to the next largest part, j − 2 to the next largest part, etc., to a
partition with j or fewer parts results in a partition with exactly j parts all distinct. From
the remaining list of positive parts {1, 2, . . . , j} we remove exactly those parts which
appear in the negative part of the partition, leaving a second partition into exactly j − k
distinct positive parts. We now have a pair of partitions (α, β), α with exactly j distinct
parts and β with exactly j − k distinct parts.
In figure 2 we illustrate this transformation performed on λ, resulting in the pair
α = (16, 13, 9, 7, 5, 4, 2, 1), β = (6, 5, 2).
It remains now to combine these two partitions in a way that recoverably preserves
their identities, and produces exactly one even part for every part in β. It is clear that we
can recover the original signed partition from this pair, and since we no longer deal here
with negative parts we will summarize the rest of the bijection. Partitions of n into distinct
parts are equinumerous with partitions into odd parts, and there are several bijections that
realize this (see the survey of Pak [9]); we use the bijection of J. J. Sylvester (which may
be found in many places, including [13]), which when applied to a partition with j parts
produces a partition into odd parts in which at least d 2j e parts are of size at least 2b 2j c + 1.
Call this γ. Any even parts in β we append (in the appropriate order) to γ; these will
all be of size less than 2b 2j c + 1. Any odd part 2m + 1 of β is added to γ in a process
that can be visualized as writing the part as 22 . . . 21 and “laying this along” γ: we add
2 to γ1 through γm , and 1 to γm+1 , this last addition producing a new even part. Since
A bijective toolkit for signed partitions
5
m ≤ d 2j e−1, the 1 is added to an odd part of size at least 2b 2j c+1, producing an even part
larger than this amount. Furthermore, all even parts produced by the process are distinct.
This process is illustrated in Figure 3. Sylvester’s bijection on α produces γ =
(19, 13, 11, 9, 3, 1, 1), with 4 parts larger than 8. From β we append parts 6 and 2, resulting in the partition (19,13,11,9,6,3,2,1,1); finally, we add the 5 from β by adding
2 to the first part, 2 to the second part, and 1 to the third part. Our final partition is
(21, 15, 12, 9, 6, 3, 2, 1, 1).
To reverse the bijection we subtract 1 from the largest even part of the partition
and 2 from all larger odd parts to produce an odd part 2m + 1 of β, continuing until the
remaining even parts are smaller than half the number t of odd parts larger than 2t; the
remaining even parts are the even parts of β. We then reconstruct the signed partition.
F IGURE 2. After subtraction.
¤
Theorem 4 is closely related to Theorem 2, sinceS
the set S2 (n), partitions of n with
no part divisible by 4, is equinumerous with the set m E(m, n), partitions of n with
distinct even parts. To see this, we simply take from largest to smallest the even parts in
an element λ of S2 (n), all of which will be twice an odd number ki ; then (k1 , . . . ) is a
partition into odd parts, which may be transformed into a partition into distinct parts. If
there are m of these, then the partition formed from collecting and properly ordering twice
F IGURE 3. The final placement of parts.
6
William J. Keith
the resultant distinct parts, and the odd parts of λ, is a new partition of n with distinct even
parts.
With this in mind, we can prove Theorem 4 bijectively by showing a correspondence
between E(m, n) and exhaustive disjoint subsets of S1 (n). With a sufficiently interesting
characterization of the latter subsets, this proof is then a refinement of the original theorem. This we do: define S1 (m, n) to be elements of S1 (n) with exactly m negative parts.
Then S1 (m, n) is equinumerous with E(m, n).
Proof of Theorem 4, refined. For α an element of S1 (m, n) with + number j, we set
l = (j − 1)/2. Write α = (β2l+1 + l, . . . , β1 + l, α−1 , . . . , α−m ) and let β be the partition (β2l+1 , . . . , β1 ), β 0 the standard conjugation of β, and βo (resp., βe ) the partition
consisting of the odd (resp. even) parts of β 0 .
The 2l + 1 parts of size at least l in α provide the parts sufficient to dominate the
negative parts, which are of size at most 2l + 1 and of which there are at most l. Once
again we will construct a partition into odd parts and from it produce a controlled number
of even parts; however, the process this time is a reflection
S of the previous.
Define for partitions λ and µ the partition π = λ µ to be the partition produced
by collecting the parts of λ and µ and reordering according to size; define the partition
π = λ + µ to have parts being the partwise sums of the parts of λ and µ, filling out the
shorter of the two with parts 0 where necessary. Note that 2l + 1 parts of size l have
the same cardinality as the triangular partition (2l, 2l − 1, . . . , 3, 2, 1). Form the new
partition δ = φ ((2l, 2l − 1, . . . , 3, 2, 1) + β), where φ is Sylvester’s bijection as applied
to a partition into distinct parts. It can be shown [10] 1 that the result of this rearrangement
of the positive parts of α is the same as beginning
with the partition (2l + 1, . . . , 2l + 1)
S
of l parts, constructing (2l + 1, . . . , 2l + 1) βo , and “laying flat” the parts in βe by twos,
in a fashion analogous to the previous proof. Viewed either way, the result is a partition
into odd parts.
We construct the target partition in E(m, n) using a variant of a map combining pairs
of partitions found in [4], for direction to which the author is grateful to Dr. Andrews.
Starting with the last (that is, most negative, largest absolute value) negative part α−m =
−(2k−m −1), subtract 2k−m −1 from the k−m -th part of δ, producing an even part which
is reordered into place. Repeat for each part α−(m−1) through α−1 . The positive parts of
δ are guaranteed to be large enough to accommodate the subtraction, and the even parts
thus produced will all be distinct. There will be exactly one of them for every negative
part in the original signed partition.
An illustration of this process applied to the positive parts of such a partition is seen
here in figures 4 through 6.
To reverse the bijection, start with the largest even part, and add 2k + 1 where k is
the unique value that will produce the k-th largest odd part in the partition.
¤
1 The
thesis of Stockhofe cited provides an excellent framework for understanding Sylvester’s bijection and
generalizes it from odd and distinct parts, understood as modulus 2, to any modulus m. Written in German, it
is available in English translation as an appendix of the doctoral thesis of the current author [8], who hopes to
make it more widely available along with journal publication of the relevant portion of his thesis.
A bijective toolkit for signed partitions
7
F IGURE 4. The signed partition α = (8, 6, 6, 5, 4, 3, 3, −1, −5). The
gray rectangles are of size j = 2l + 1 by l, comprising the required
parts of the positive portion of the partition and containing the possible
negative parts (the striped region of length 2 is forbidden due to strict
inequality).
F IGURE 5. α with the positive parts rearranged into odd parts (reading
the positive portion of the diagram by columns).
F IGURE 6. The subtraction process for Theorem 4, with the diagram
being read by columns.
Combined with the previous proof, this proof defines a bijection between the sets of
signed partitions S1 (m, n) and S(m, n). It might be instructive to produce a more direct
map.
8
William J. Keith
Pausing for a moment of analysis, let us observe the process which produced the
even parts in each proof. In the first theorem, the number of even parts to be produced was
equal to the excess of the number of positive parts over negative parts; we used negative
parts to “cancel” elements from a list drawn directly from the positive portion, and each of
the remaining positive parts was added to produce an even part. In the second theorem, the
number of even parts to be produced was equal to the number of negative parts themselves,
and the production was accomplished by direct subtraction from sufficiently large odd
parts. If the subtraction of negative parts is to be a feature of bijective proofs for identities
with signed partitions and one behavior or the other holds in the identity, this suggests we
might look at the corresponding technique for proof.
Bearing this in mind, let us turn to Theorem 3. As Andrews points out in [2], this
theorem is a variation of the first Göllnitz–Gordon identity. That theorem is implied by
a more general theorem ( [5], Theorem 7.11), for which the relevant specific case states
that partitions into parts ≡ 1, 4, or 7 (mod 8) are equinumerous with partitions in the
class D2,2 , defined as those partitions (b1 , . . . , bs ) for which bj − bj+1 ≥ 2 if bj is
odd, bj − bj+1 > 2 if bj is even, and at most 1 part is 1 or 2. (The latter clause is, of
course, implied by the former; it is retained for consistency with a more general statement
discussed later.) Hand calculation of the two types of partition is easy for small n, and
swiftly leads to the conjecture that signed partitions of type G1 (n) with excess of positive
parts over negative parts m occur equally often with partitions in class D2,2 with exactly
m even parts, or, that those with exactly m negative parts are equinumerous with those
having m odd parts. The proof using the technique above is now almost instant.
Proof of Theorem 3, refined. In a signed partition α from G1 (n), the required positive
portion is of size 2j by j, or a square of 2s. Rearrange this square into a list of parts
(2 · 1, 2 · 3, . . . , 2 · (2j − 1)), and add those parts above. We get a partition β into exactly
j distinct even parts, say (β1 , . . . , βj ). For each negative part −(2k1 + 1), subtract 2 from
β1 through βki and 1 from βki +1 , producing an odd part at the end.
The process is illustrated in figure 7. The resulting partition is clearly of class D2,2 ,
and by reversing the prescribed recipe for subtracting parts, incrementing each even part
by 1 and adding 2 to larger parts, we reverse the bijection. Each negative part produces an
odd part in the ordinary partition, and the number of even parts is precisely the excess of
the + number over the − number.
¤
(We could also understand this map in the following way: split the square into two
triangular halves, subtracting 1 from α1 , 3 from α2 , etc., and 2j − 1 from αj , leaving
a partition β into exactly j distinct odd parts, say (β1 , . . . , βj ). From the set of magnitudes subtracted, that is, {1, 3, . . . , 2j − 1}, remove any negative parts which appear in α.
Suppose that the list remaining is γ = {2k1 + 1, . . . , 2km + 1}. Add these parts to β by
“laying flat”: for each part 2ki + 1 of γ, add 2 to β1 through βki and 1 to βki +1 , producing
an even part.)
This proof is so simple that it lends itself to expansion. On the one hand, it would be
nice to be able to prove the entire family of identities alluded to in the paragraph before the
proof, which in general state that for 0 < i ≤ k partitions of n into parts 6≡ 2 (mod 4), and
A bijective toolkit for signed partitions
9
F IGURE 7. The bijection for Theorem 3, applied to α =
(14, 10, 10, 8, −1, −3, −7). The partition splits into the pair β =
(13, 7, 5, 1), γ = (5). The pair combines to form the target partition
(15, 9, 6, 1).
6≡ 0, ±2i − 1 (mod 4k), are equinumerous with partitions of the class Dk,i (n), partitions
with odd parts, distinct, at most i − 1 parts of size 1 or 2, and differences among parts
bj − bj−k+1 ≥ 2 if bj is odd and > 2 if bj is even. The first example would be the case
k = i + 1 = 2, which relates to the second of the Göllnitz–Gordon identities. The same
hypotheses define D2,1 as define D2,2 save that no parts can be 1 or 2. From our bijection,
it is easy to define the class of signed partitions that gives rise to this set: they are simply
those signed partitions with the same hypotheses as in the theorem, save that the even
positive parts are strictly larger than twice the + number. The case for modulus k > 2
presents a different appearance. A more complete exploration we undertake in the next
section, after proving the fourth of the theorems we are beginning with.
The simplicity of this proof also makes it a good showcase for an illuminating principle: there can be more than one way to combine the positive and negative parts of a
signed partition, each giving rise to different classes of partitions, or the same classes
with different statistics preserved. For example, in the theorem above we could take the
positive parts and conjugate their 2-modular diagram (that is, divide all parts by 2, conjugate the resulting partition, and multiply the conjugate by 2 again). This partition has
at least j largest parts of size exactly 2j. From this partition we can reversibly perform a
direct subtraction of the negative parts, removing −(2ki − 1) by subtracting 2ki − 1 from
part ki , and reordering the resulting odd part into place. The resulting ordinary partition
of n has distinct odd parts, of a number equal exactly to the number of negative parts in
the original signed partition of n, and the class is further defined by the restriction that,
when the largest odd part is 2m + 1, either all parts from {1, 3, . . . , 2m + 1} appear or
10
William J. Keith
(exclusively of the previous) the largest part is even, and in the latter case appears at least
a number of times equal the number of parts from {1, 3, . . . , 2m+1} not appearing. More
such classes can be produced; all are immediately equinumerous.
Before we head off in new directions, let us conclude our tour of Andrews’ paper
with his Theorem 10, number 5 in our listing. It may be noticed that the positive parts of
the signed partitions in this theorem are exactly the same as those of Theorem 4, while
the negative parts would fit the hypotheses of that theorem if doubled and increased by
1. On the other hand, the Rogers-Ramanujan identities are consequences of the Göllnitz–
Gordon theorem, so we might have expected a closer relationship between this set and the
signed partitions of Theorem 3. It may be illuminating to consider this as we follow the
proof.
Proof of Theorem 5, refined. Once again, we employ the theorem mentioned by Andrews.
The first Rogers-Ramanujan identity equates the number of partitions of n into parts ≡
1, 4 (mod 5) with partitionsS
of n into parts differing by at least 2. The set R2 (n) consists
of those partitions α = αo αe into odd and even parts (notation as before), where the
parts of αe are just twice parts ≡ 1, 4 (mod 5) and are thus equal in number to partitions of
|αe | into even parts differing by at least 4. Bearing this identity in mind, we can prove the
equivalence of the theorem’s two classes of partitions by showing equivalence between
the signed partitions described, and partitions in which any even parts differ by at least 4.
We would like, therefore, to produce ordinary partitions in which the even parts
are under control. A handy way to do this would be to start with a partition entirely
into odd parts, and so we manipulate the positive parts of the partition the same way
we did for Theorem 4. If the + number is j = 2l + 1, and we call
S the positive portion
(l + β1 , . . . , l + β2l+1 ) with βi possibly zero, split β 0 into βo βe by odd and even
parts, and note that S
these parts
¡ are all
¢ at most 2l + 1 or 2l respectively. Construct γ =
(2l + 1, . . . , 2l + 1) βo + 2 βe 0 /2 , where by the last term we mean adding βe by twos
laying its 2-modular diagram along the previous partition. (See the earlier proof for an
illustration of the process.) The result is a partition into odd parts.
To construct even parts out of this partition we need to subtract odd parts, and are
presently equipped with negative parts not odd but distinct. The solution is, of course,
to employ Sylvester’s bijection once more, turning our distinct negative parts into odd
ones. Since the largest of these is of size at most l, N.J. Fine proved analytically ([5],
p. 26), and Christine Bessenrodt showed ([6]) that Sylvester’s bijection gives, that the
resulting partitions into odd parts have their largest part plus twice the number of parts at
most 2l + 1. Aesthetics suggest that our bijection will make maximum use of the leeway
granted us in subtraction, and so we do. The map is the same one used in our proof of
Theorem 4.
Suppose that the transformed negative side of the partition is now δ = (δ1 , . . . , δs ).
For each 1 ≤ i ≤ s, say δi = 2di + 1. Set 2ji − 1 = δi + 2(s − i). Starting in order with
i = 1, subtract 2ji − 1 from part γji of our partition into positive odd parts, then reorder
the parts. Next add a compensatory 2 to parts γ1 through γs−i . Since s − i + 1 ≤ l and
γs−i+1 ≥ 2l + 1 > 2l − 1 ≥ 2ji − 1, the subtraction produces an even part of size at
least 2 to begin with. Since ji is strictly decreasing as i approaches s, the compensatory
A bijective toolkit for signed partitions
11
increment added to the larger parts of γ means after each step the next even part produced
will differ from the previous by at least 4.
We can recover γ and δ by reversing the procedure on such a partition, starting with
the largest even part. We subtract 2 from the largest 0, 1, 2, etc. parts, then add the unique
2ji − 1 = 2di + 1 + 2(s − i) that produces from the largest remaining even part a new
odd part which reorders into the ji -th place.
There are numerous statistics preserved by Sylvester’s bijection which have been
well-studied, and which can be translated into further descriptions of the behaviors of the
even parts of the partitions produced here. For a concise summary, the reader is referred
to [13].
¤
Once again, a note on a related identity similar to our construction of the second
Göllnitz–Gordon identity: the second Rogers-Ramanujan identity equates partitions in
which differences between parts are at least 2, and the smallest part is greater than 1,
with partitions into parts ≡ 2, 3 (mod 5). The related family can be constructed as in the
above simply by insisting that no 2 ever be produced; for this we require that we do not
simultaneously have both that 2j1 + 1 = 2d1 + 1 + 2(s − 1) = 2l − 1, and that γl = 2l + 1.
In turn, these do occur if in the original signed partition the largest distinct negative part
was of size −l and no part of size 2l appears in βe , i.e., the next-to-smallest positive part
of α equals the smallest positive part.
Thus, partitions of n in which the even parts are ≡ 4, 6 (mod 10) are equal in number to signed partitions of n in which there are an odd number j = 2l + 1 of positive parts,
all of size at least l, negative parts are distinct and all ≤ l, and a negative part of size −l
occurs only if the smallest positive part of α is distinct from the next-smallest.
3. New Directions
While we have easily obtained a few refinements and some of the identities most closely
related to our starting point, the field in which we are working now seems to have a much
larger range of results in easy reach.
3.1. Partitions of (m, c)-type
Let us begin by generalizing the modulus for the identities: each of the theorems described
so far deals with partitions into odd and distinct parts, features apparent in the 2-modular
diagram of a partition. For example, wherever a bijection calls for splitting a guaranteed
amount of 2s into a triangle of odd parts, we can split a guaranteed amount of moduli m
into parts congruent to c and m − c modulo m, and perform analogous manipulations.
Such partitions are said to be of (m,c)-type, and are discussed in [13].
Theorem 2 does not seem, immediately, to yield to this sort of manipulation. Our
signed partition there concerns distinct parts, not odd ones, and indeed, the bijection calls
for carving our required positive parts into two triangles. On the other hand, Theorem 4
is much more adaptable to this viewpoint. The (m, c) version of that theorem (the refined
version our bijection proved) is as follows:
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William J. Keith
Theorem 6. For 0 < c < m, let S1,(m,c) (k, n) denote the number of signed partitions
of n with + number j ≡ c (mod m), with λi − λi+1 > 0 only if i ≡ 0, c (mod m),
λj > |j|m , and k distinct negative parts ≡ −c (mod m), of size < j. (Equivalently, by
taking conjugates, we require that the largest positive part be ≡ c (mod m), that it be
repeated at least as often as its m-weight, and all positive parts are ≡ 0, c (mod m).) Let
S2,(m,c) (k, n) be the number of ordinary partitions of n into parts ≡ 0, c (mod m) with
exactly k parts divisible by m, all distinct. Then S1,(m,c) (k, n) = S2,(m,c) (k, n).
Proof. Run through the bijection used to prove Theorem 4, using m-modular diagrams
instead of 2-modular diagrams. The positive parts of a partition in S1,(m,c) (k, n), read by
columns, are all of residue 0 or c modulo m. Replace the 2s in that bijection by m, and
the 1s by c. We have a required square of ms of size j. Move positive parts divisible by m
to lay flat on the top of the square, and subtract off negative parts −(tm − (m − c)) from
the tth positive part thus produced. A partition in S2,(m,c) (k, n) results.
¤
We could, again, take the partition formed by the distinct parts of the target partition
divisible by m, and perform Sylvester’s bijection upon their m-weights to obtain a list
of parts not divisible by 2m, a class closer to that of the original theorem. We could also
note, as we did before, that this class is of the same type as E(m, n) of Theorem 2. While
it is not immediately obvious what the signed partitions should be, we should be able
to reverse the direction of the bijection given in the proof of that theorem, replacing odd
parts with parts ≡ c (mod m), to find out what such a class should be. Observe particularly
Figure 3, remembering to read horizontally.
The parts divisible by m would have been produced by either interleaving a part
divisible by m, or “laying flat” a part ≡ m − c (mod m), either fitting to the side of or
beneath the “m-modular Durfee square” – the visual square2 of size k in the m-modular
diagram of a partition formed by moduli m when the first k parts are of size at least km
– in a partition with parts all ≡ c (mod m). In the latter case the addition of c and m − c
would produce a part divisible by m. Such a partition into parts ≡ c (mod m) would
have arisen after applying the m-modular generalization Lm ◦ L1 of Sylvester’s bijection
found in [10] (or the author’s English translation thereof, footnoted earlier) to a partition
whose conjugate contained all parts ≡ 0, c (mod m) of consecutive m-weights up to the
maximum appearing: i.e., parts c, m, c + m, 2m, c + 2m, . . . , c + (k − 1)m OR km.
We are finally closing in on the starting signed partition. A partition of the type
described above arises when we remove from a series of large, equal parts the (m, c)
equivalent of a triangle: instead of alternating even and odd parts 1, 2, 3, . . . we remove
parts m − c, m, 2m − c, 2m, 3m − c, . . . from the list of required parts, leaving parts (in
reverse order) c, m, c + m, 2m, . . . . The sum of such parts in order is m, 2m, 3m, . . . .
There are various ways to realize this list of required parts: we may require that the positive parts differ by at least m, the smallest being of size at least m, or we may describe it,
in the spirit of our original theorem, in terms of the + number. With the latter approach,
we have built the following generalization:
2 This
is a simplified definition applicable to partitions in which the nonzero residues modulo m are all equal.
A bijective toolkit for signed partitions
13
Theorem 7. Let E(m,c) (k, n) denote the number of ordinary partitions of n into parts
≡ 0, c (mod m), with exactly k parts divisible by m, none of which is repeated. Let
S(m,c) (k, n) denote the number of signed partitions of n with + number j, k the excess
of + number over − number, in which each positive part is at least as large as md 2j e,
and if j is even the first 2j parts are at least m(d 2j e + 1); first differences λa − λa+1 are
0 among positive parts unless a ≡ 0, c (mod m), exception at least 1 at a = 2j if j even;
and negative parts are ≡ 0, m − c (mod m), distinct, and not exceeding mj
2 (if j even) or
m − c + mb 2j c (if j odd). Then E(m,c) (k, n) = S(m,c) (k, n).
When seeing that this theorem really is a general case of Theorem 2, it should be
noted that in that theorem m = 2, so c = 1 = m − c and all parts are automatically
≡ 0, c (mod m). Instead of requiring m = 2 be added to the first 2j parts of a partition
with an even j, that theorem adds 1 to each of the required parts.
The other theorems may be generalized similarly. Indeed, in some cases the (m, c)
variants are so similar in structure to the original theorem that, as Zeng notes, the resulting generating function identities are algebraically equivalent to the originals with a
substitution of variables. Additional richness can be ensured by considering instead, e.g.,
m-falling or m-rising partitions, in which the (least positive) residues of parts mod m are
allowed to vary but must monotonically rise or fall.
3.2. Modulus 3
Heading in other directions, we quickly come across the realization that residues for larger
moduli can be difficult to control when combining positive and negative parts of a signed
partition – for m = 2 all such problems are swept under the rug since there is only one
possible nonzero residue, which when added to itself yields m. The modulus m = 3,
however, has a richer structure while still being controllable in convenient ways, some of
which turn out to reflect interestingly on theorems present in the prior literature. Let us
begin with a generalization of Theorem 3, in which rather than the (m, c) generalization
we accept any residue on the negative parts:
Theorem 8. The number of signed partitions α of n in which all positive parts are divisible by 3, each at least 3 times the + number, with negative parts not divisible by 3, all less
than 3 times the + number and having distinct 3-weights, equals the number of ordinary
partitions β of n in which all parts have different 3-weight, and the difference is greater
than 1 if the larger of the two is divisible by 3.
Proof. Note the slight recharacterization of distinctness required for the negative parts.
When our modulus was 2, distinct odd parts were automatically of distinct 2-weight.
Here we would disallow, say, parts -1 and -2, or -4 and -5. On the other hand, parts may
differ by as little as 2, when we have, e.g., parts -2 and -4, so this condition is different
from simply requiring that the negative parts differ by 3.
Because the 3-weights of all negative parts differ, the proof used for Theorem 3
is adaptable in its entirety. We also obtain the refinements that the number of parts not
divisible by 3, and divisible by 3, are equal to the number of negative parts and the excess
of positive over negative parts, respectively.
¤
14
William J. Keith
A brief observation shows that this theorem holds with all instances of 3 replaced by
any modulus. The 3-modular case is of particular interest, however, because the partitions
involved are closely related to those in a theorem of Issai Schur. These were expanded
upon by K. Alladi, Andrews, and finally Ae Ja Yee in a succession of papers we will
follow (Andrews’ paper [3] is the start of this series, and is a good reference for the
original theorem of Schur; the final paper in the sequence of interest here is [11]). These
papers prove the identity of those sets with a number of others, combinatorially and by
generating function manipulations. The original theorem is:
Theorem 9 (Schur’s Theorem). Let A(n) denote the number of partitions of n into parts
congruent to ±1 (mod 6). Let B(n) denote partitions of n into distinct nonmultiples of 3.
Let D(n) denote partitions of n in which the differences between consecutive parts are
at least 3, and strictly greater than 3 if 3 divides the larger part. Then A(n) = B(n) =
D(n).
Andrews and Alladi ([1], and see [3]) add a family of type C, partitions into odd
parts appearing not more than twice, equivalent to those above. The paper [3] proves
these equivalencies using short generating function manipulations, though it skips any
proof at all that these three are equivalent to those of type D. Partitions of type B and
C can be obtained bijectively from those of type
Glaisher-style
rewrites: map
P A through
P
a partition in which a part 6k + 1, 5 appears
bik 2i (resp.
cik 3i ) times, bik = 0, 1
(resp. cik = 0, 1, 2) to that partition in which each part 2i (6k + 1, 5) appears bik times,
i.e. partitions of type B (resp. each part 3i (6k + 1, 5) appears cik times, partitions of type
C).
The proof that these three classes are equivalent to type D is a much trickier affair
from the viewpoint of q-series. However, we can add another equivalent family, of signed
partitions, which can be rapidly mapped to those of types D and B, establishing this part
of the equality.
The partitions of D(n) are a subset of the ordinary partitions discussed in Theorem 8. The excess partitions are exactly those in which there are two consecutive parts
(. . . , 3l + 4, 3l + 2, . . . ), which have different 3-weight but differ only by 2. Observe
the signed partition from which such an ordinary partition arises in our proof: this occurs precisely when we subtracted from β negative parts of some size −(3ki + 2) and
−(3(ki + 1) + 1), and there was not a column of 3s of length ki + 1 lying above the
required square of 3s in the signed partition. If we select only those signed partitions in
which such a column exists whenever we have successive negative parts of residue 2 followed immediately by 1, we obtain a class of signed partitions equinumerous with those
of type D. Call such partitions of type F, and let F (n) enumerate those that partition n.
Theorem 10. B(n) = F (n) = D(n).
Proof. Partitions of type B can be produced from those of type F using a 3-modular
variant of the proof technique we described just after proving Theorem 3. Take the square
of required 3s and split it into a triangle of parts all 1 (mod 3), and an opposing triangle of
parts all 2 (mod 3); list these. If the + number is j, all nonmultiples of 3 from 1 to 3j − 1
appear. We also have a list of multiples of 3 consisting of the excesses of the positive
A bijective toolkit for signed partitions
15
parts over the required minimum. Read this list of excesses vertically to construct a list of
positive parts divisible by 3, of size at most 3j.
Remove from the list of nonmultiples of 3 any negative parts which appear. We
remove at most 1 part of any given 3-weight, so the differences among the remaining
parts are always ≤ 2 unless parts 3ki + 2 and 3(ki + 1) + 1 were removed; but in this case
we defined our family of signed partitions so that one of the “excess” columns is exactly
of size 3ki . We use the 3s of this column by adding 3 to the nonmultiple 3ki + 1 and each
smaller part as far as we have 3s to add. The row of 3s thus produced will always be of
length at least j + 1, and will not reach further than the last nonmultiple. The remaining
excess columns of 3s from the positive part of our partition are all of size ≤ j; we add
them unitwise in the same fashion, after completing the subtraction process above.
We reverse the map by subtracting off rows of 3 unitwise until the rows get too long
to have been excess 3s over the number of positive parts that could have produced our
nonmultiples, given the subtractions that had to occur. Replace these missing nonmultiples, removing columns of 3s when necessary to add in −(3ki + 2) and −(3(ki + 1) + 1).
Recombine the full list into a square and add the columns of 3s obtained in previous
stages, listing the negative parts removed.
¤
In [3] Andrews adds, and in [11] Yee proves bijectively the equivalence of, a fifth
class of partitions E(n) related to a family of orthogonal polynomials. The partitions of
E(n) are those in which odd parts are distinct, even parts may appear at most twice, the
difference between two parts is never 1, is 2 only if both are odd, and partitions with c
c
pairs of repeated even parts are counted witth weight (−1) . The partitions of type D are
a subset of type E, and Yee produces a sign-reversing involution that fixes D and pairs
off all other elements. It was hoped in this paper that the bijection might be simplified by
considering the new class of signed partitions added to the group, but as of submission this
has not been a successful search. A future investigator might find the challenge interesting.
3.3. Conjugation
We have been using a new version of the useful old Ferrers diagram to illustrate signed
partitions. Our presentation has served well in the construction of various bijections, and
has the advantage that parts are listed vertically in descending order, including negatives.
However, it may not be ideal for all purposes. One glaring feature it lacks is the phenomenally useful operation of conjugation, easily understood for the standard Ferrers diagram
as reflection about the main diagonal in the plane. Any attempt to produce useful bijective
tools for investigating signed partitions must consider some version of a generalization of
this map.
With the graphical presentation we have been using, reflection across either diagonal
does not result in the diagram of a signed partition. The only linear transformation of the
plane that does guarantee a new signed partition diagram is rotation about the origin by a
half revolution. Numerically, this has the effect of mapping
(λ1 , . . . , λj , λ−1 , . . . , λ−k ) → (−λ−k , . . . , −λ−1 , −λj , . . . , −λ1 )
i.e., reversing the order of parts and negating all.
16
William J. Keith
Perhaps interestingly, this operation commutes with standard conjugation on ordinary partitions under the following map. Take an ordinary partition λ = (λ1 , . . . , λj )
and denote the standard conjugation of λ by λ0 . Suppose λ has (standard, or 1-modular)
Durfee square of size s, i.e. s is the largest integer for which λs ≥ s. Then the Frobenius
symbol of λ is given by
¶
µ
λ1 − 1 λ2 − 2 . . . λ s − s
.
λ0 1 − 1 λ0 2 − 2 . . . λ 0 s − s s
Each row is a partition (with possibly one zero part) into distinct parts. Standard
conjugation on ordinary partitions switches the rows. If we add 1 to each entry and negate
the bottom row, then list all elements of the Frobenius symbol as a signed partition, we
obtain
λ → (λ1 , λ2 − 1, . . . , λs − s + 1, −(λ0 s − s + 1), . . . , −(λ0 1 )) .
This is a map between ordinary partitions of n with Durfee square of size s and
the set of signed partitions in which the sum of the absolute value of all parts is n + s,
with exactly s positive and s negative parts, each distinct. (Sign matters to distinctness; we
allow parts j and −j.) Rotation of the resulting signed partition about the origin commutes
with conjugation of the original ordinary partition.
(A remark: if we added 12 instead of 1, the sum of the absolute values of the halfinteger parts thus produced would have been n regardless of s. The resulting vectors are
apparently of interest in statistical physics concerning the distribution of energies among
particle-antiparticle pairs; I am not expert in this field, but mention the connection for
those whose curiosity might be piqued by related investigations.)
Alternatively, we may seek a different diagram more suitable for a conjugation operation. Treating the positive and negative parts as separate partitions, and conjugating
each, results in a new signed partition. This operation reduces to regular conjugation for
ordinary partitions viewed as signed partitions with no negative parts. It is a linear transformation on the plane if we use a different diagram, thus:
In the top of Figure 8, we have illustrated the signed partition
(8, 7, 7, 6, 5, 4, 4, 3, −1, −3, −4, −7, −8)
using a version of the Ferrers diagram in the fourth and second quadrants, with the negative parts in the second. While the parts are no longer visually arranged in descending
order of magnitude, we can generalize conjugation visually. In the bottom left of the figure, we have reflected the diagram about the diagonal y = −x. The resulting partition
is (8, 8, 8, 7, 5, 4, 3, 1, −1, −2, −2, −2, −3, −4, −4, −5), the partition wherein the positive and negative parts were each conjugated separately. This diagram and operation thus
extend both the standard Ferrers diagram and the operation of conjugation to ordinary
partitions as the special case of signed partitions, with no negative parts.
In the bottom right of the figure, we employ the third transformation of the plane
that preserves the double cone, reflection across y = x. It could also be understood as
composing the previous two maps, rotation followed by separate conjugation, since these
three maps are the ball game as concerns possible transformations. Separate conjugation
A bijective toolkit for signed partitions
F IGURE
8. Two
versions
of
conjugation
(8, 7, 7, 6, 5, 4, 4, 3, −1, −3, −4, −7, −8).
17
applied
to
is weight-preserving on signed partitions, but this operation, like rotation, exchanges a
signed partition of n for one of −n.
3.3.1. Partitions of Zero. A somewhat astounding tangent arose at this point in investigations. In the usual study of partitions, there is only one partition of zero, the empty set.
Any signed partition in which the positive part partitions n and the negative part partitions
−n is a partition of zero, and it will remain so under the last reflection operation described
above.
2
Call such a signed partition of height n; there are p(n) such partitions, p(n) being
the number of ordinary partitions of n. Being a partition of zero is a property preserved
under many more transformations of the plane than weight in general; for example, multiplying all parts by c preserves the property.
An early stab at illustrating such partitions represented them as convex paths over
the x axis:
18
William J. Keith
F IGURE 9. Three partitions of 0, of height 11, illustrated as convex
polygons on the axis.
j
X
λi , j)
(0, 0) → (λ1 , 1) → (λ1 + λ2 , 2) → · · · → (
i=1
j
X
λi − λ−1 , j + 1) → · · · → (0, j + k) . (4)
→(
i=1
These vary from “thinnest” at (n, −n) to “thickest” at (1, . . . , 1, −1, . . . , −1). The
author became curious as to how the areas were distributed between their minimum of
n and their maximum of n2 . In particular, it seemed natural to ask how many partitions
there were of a given area at all. The following table lists the number of partitions of zero
with a given area, up to an area of 50.
A bijective toolkit for signed partitions
19
Area
Pop
0
1
1
1
2
1
3
3
4
2
5
3
6
5
7
4
8
6
Area
Pop
9
10
10
6
11
10
12
16
13
10
14
15
15
24
16
16
17
25
Area
Pop
18
31
19
26
20
33
21
46
22
38
23
49
24
62
25
55
26
66
Area
Pop
27
79
28
72
29
96
30
107
31
96
32
121
33 34
141 131
35
159
Area 36
Pop 185
37
173
38
199
39 40
237 216
41
266
42 43
296 272
44
334
Area 45
Pop 376
46
349
47
417
48 49
465 448
50
512
The grouping is deliberate: observe the behavior of the areas by threes. Areas of size
3k + 1 are deficient in number compared to their two neighbors; areas of size 3k appear
more often, and areas of size 3k + 2 even more, roughly as often as areas of size 3k + 4,
one grouping up.
Why?
The areas under each half of these convex paths are half-integers that are halves of
squares and truncated squares; the generating function for the area under any such path is
X
z
Q
h,k≥0
1≤j≤h
1≤l≤k
(1 − z
h2 +2hk−k2
2
h2
2
(h−j)2
−
2
q h−k
+jk q j )(1
l2
− z − 2 q −l )
When k = 0 we get the analogous expression for ordinary partitions. Curious
whether this phenomenon held for those ordinary partitions, calculations were made, but
the half-integer weights for such paths grow slowly and a pattern was difficult to discern
in early data. However, a simpler approach yielded similar behavior; consider the “second
integral” of a partition to be the progressive sum λ1 + (λ1 + λ2 ) + (λ1 + λ2 + λ3 ) + . . . ,
or λ1 · k + λ2 · (k − 1) + · · · + λk · 1. (”Groups of people arrive in bunches, most at first
and then fewer later. You have to put up the lot of them at a local hotel until they have
all arrived. How many person-nights are you going to be stuck for?”) The name “second
integral” is chosen as the inverse of the much-studied first differences, second differences,
etc. of a partition, which are the analogues of differentiation for the partition viewed as a
discrete, nonincreasing function on the natural numbers. (The first integral is its weight.)
The total area thus produced can range from n to (n2 + n)/2. The table below gives the
number of times areas arise:
20
William J. Keith
Area
Pop
0
1
1
1
2
1
3
2
4
1
5
2
6
3
7
2
8
2
Area
Pop
9
4
10
3
11
4
12
5
13
3
14
5
15
7
16
4
17
7
Area 18
Pop 8
19
6
20
8
21
11
22
7
23
9
24
13
25
9
26
11
Area 27
Pop 16
28
12
29
15
30
18
31
13
32
17
33
20
34
17
35
21
Area 36
Pop 24
37
19
38
24
39
30
40
22
41
28
42
34
43
26
44
34
Area 45 46 47 48 49 50
Pop 38 30 37 43 37 42
The growth is slower, but exhibits similar behavior; indeed, 3k, 3k+2, and 3k+4 are
now all roughly equal. Again, one would like to know why, and if the behaviors described
persist indefinitely, persist indefinitely with exceptions, or later yield to domination by the
general increase.
This entire line of questioning has rather departed from anything involving signed
partitions. However, since it arose due to this investigation, it is presented here as an item
of interest rooted in Andrews’ happy revisiting of Euler’s overlooked partitions.
3.4. Profiles and Complements
One valuable feature of the standard Ferrers diagram is the profile of a partition, the outer
edge of the squares occupied by dots in its Ferrers diagram. If the partition has fewer
than N parts, all of size less than M , it fits inside an N × M rectangle. A lattice path
from the lower left corner of that rectangle (the point (0, −N ) if we consider the standard
Ferrers diagram as being in the fourth quadrant) to the upper right corner (the point (M, 0)
likewise) is a set of steps north and east in this rectangle, and every such path describes
a partition with a given profile. The statistics of the partitions thus described, especially
the weight, form a set of useful statistics on lattice paths. The number of partitions in the
N × M rectangle with weight j is the coefficient of q j in the q-binomial coefficient
·
N +M
M
¸
=
q
(q; q)N +M
,
(q; q)N (q; q)M
so
these
´ objects are analogues of the usual binomial coefficients, i.e.
³
h named
i because
N +M
N +M
→
as q → 1.
M
M
q
Conjugation of partitions is a bijection between lattice paths in the N × M rectangle
and the M × N rectangle. It reverses the order of all steps, then exchanges east for north
steps and vice versa. The weight of the path is preserved but not, we see, its dimensions.
A bijective toolkit for signed partitions
21
One involution on lattice paths in the N × M rectangle is the complement, in which we
simply reverse the order of steps, or equivalently, rotate the rectangle a half turn and take
the other side of the path as our new partition. This preserves dimensions
ibut not (usually)
h
N +M
weight. Indeed, it shows that the coefficients of the polynomial
are symmetric
M
q
about q N M/2 , an analytic identity that can be rendered
·
¸
·
¸
N +M
N +M
= qN M
.
M
M
q
q −1
(That latter base seems to suggest that any proof featuring this identity is ripe for
a new look involving signed partitions. We can combinatorially interpret the above as
producing partitions in the N × M rectangle by subtracting from an N × M required
positive block a partition into negative parts that themselves fit in such a rectangle.)
On the other hand, consider our original Ferrers diagram. A signed partition, say
(λ1 , . . . , λj , λ−1 , . . . , λ−k ), displayed in such a fashion has a profile constituting a lattice
path from (−λ−k , −k) to (λ1 , j), necessarily starting and ending with two north steps,
and passing through the origin via two east steps unless one or both of the sides are
empty. By removing those steps, which corresponds to removing the largest “hook” of the
positive and negative portions of the partition, we obtain a correspondence between lattice
paths in the (N − 2)x(M − 2) box and several signed partitions with exactly M parts,
at least some negative and some positive, for which λ1 + λ−k = N . The correspondence
is not a bijection – the east steps surrounding the origin can be placed on any node (i.e.,
separate the parts before and after using the two steps), meaning that each such path maps
to exactly N + M − 3 signed partitions of the type described in the N × M rectangle.
Unfortunately, the weight changes under this map in a fashion very dependent on each
path, so it does not make a convenient way to count signed partitions in the N × M box.
Instead, visualize the following: suppose we have a signed partition into at most M
total parts, with at least one part negative and λ1 + λ−k ≤ N (λ1 = 0 if no parts are
positive). Diagram it with our original Ferrers diagram. Up-justify the positive parts so
that the height difference from top to bottom is exactly M ; draw a vertical line from the
origin if no parts are positive. We have the profile of a partition in the N × M box, with at
most M −1 parts. The weight of our signed partition is exactly the weight of this partition,
less a negative rectangle of size M λ−k .
Thus, call p± (N, M ; n) the number of signed partitions in the N ×M box of weight
n. We then have
N
M
X
n=−N M
·
p± (N, M ; n)q n =
N +M
M
¸
+
q
N
X
m=1
·
q −M m
N +M −1
M −1
¸
.
q
The map is illustrated in figure 10.
It is obvious that, if λ is an ordinary partition n, the weight of λ will be the same
as its complement, say c(λ), in any N × M rectangle that λ fits in with N M = 2n. It is
of interest to study what partitions might be called “self-complementary,” but as studied
in the previous literature this characteristic is defined with reference to a given N × M
22
William J. Keith
F IGURE 10. A signed partition mapped to an ordinary partition in the
21x13 box.
h
i
+M
box, a choice external to the partition itself. Stanley showed that NM
, evaluated at
q = −1, gives the number of self-complementary partitions in the N × M rectangle.
(Note: if a partition is to be self-complementary in any rectangle at all, it will be
so in a rectangle of height equal to the number of parts, and length equal to the largest
part plus the smallest part, or by conjugation of length equal to λ1 and height equal to λk
plus the number of times λ1 is repeated. Suppose we simply ask how many partitions of
n have the property that k(λ1 + λk ) = 2n or the conjugate box described has area 2n,
and call these partitions complementable. The first complementable partition that is not
self-complementary in the associated box occurs for n = 15. It is far too tangential to this
paper to discuss here, but might be another question for research.)
The relevance of this discussion in this paper is that a simple variant of our basic
complementation transformation always preserves the weight of a signed partition!
Namely, let λ = (λ1 , . . . , λj , λ−1 , . . . , λ−k ) be a signed partition, and let N =
max(λ1 , −(λ−k )), M = max(j, k). Then both the positive parts and the negative parts
are partitions that fit in an N ×M or larger rectangle. Take the complement of the positive
parts in the N × M rectangle, and reverse the sign of all parts to produce negative parts
γ−1 , . . . , γ−m . Take the complement of the negative parts in the same size rectangle, and
again reverse the sign of the resulting parts to create positive parts γ1 , . . . , γl . Gather these
as a new signed partition γ.
Suppose the weight of the positive parts of λ totals A, and the weight of the negative
parts totals −B. Then |λ| = A−B, and |γ| = (−N M +A)+(N M −B) = |λ|. A signed
partition is self-complementary under this map if the positive and negative portions are
themselves complements of each other in some rectangle, which is naturally the rectangle
to be chosen. Self-complementary partitions of 0 of height n are, then, given exactly by
complementable partitions of n.
3.5. Issues for Exploration
Any newly formalized field of inquiry has an embarrassment of riches when it comes to
open questions. Some of the more interesting questions I have come across in preparing
A bijective toolkit for signed partitions
23
F IGURE 11. A complementation in the 5 × 5 rectangle; the weight,
n = −4, is preserved.
this paper, which might be worth an investigator’s time, include the following tangents to
questions discussed above:
• When is a class of signed partitions a “natural” intermediary between two classes of
ordinary partitions?
• What combinatorially interesting objects are enumerated naturally by classes of
signed partitions? A good example is conjugacy classes in SL(n, Z); the Jordan
form of the matrices will have blocks of lengths partitioning n, each having 1 or -1
on the main diagonal; thus, these are described by signed partitions wherein the sum
of the absolute value of parts is n.
• How many complementable and self-complementary signed and ordinary partitions
of n are there? Provide some nice generating functions for these and describe how
the two sets differ.
• Explain the behavior of partitions of zero in their arithmetic progressions modulo 3
and 6.
• Explore the connection between lattice paths in the (N − 2)x(M − 2) rectangle,
and the signed partitions in the N × M box to which they map under the possible
placements of the origin.
• Partitions of type B, with size restricted, are addressed by the first Borwein Conjecture. Could our map, or a similar one, make headway toward this open question?
The author wishes to thank Dr. Andrews for reference to a number of papers which
added both useful resources and several useful questions this paper could tackle. As the
list of issues above shows, there is much more to be explored in nearby regions of this
new way of looking at partitions.
References
[1] Alladi, K; Andrews, G.E.; and Gordon, B.: Generalizations and refinements of a partition
theorem of Gollnitz. J. Reine Angew. Math. 460:165-188 (1995)
[2] Andrews, G.E. Euler’s “De Partitio Numerorum.” Bull. of the Amer. Math. Soc. 44(4):561-573
(2007)
24
William J. Keith
[3] Andrews, G.E. Schur’s theorem, partitions with odd parts and the Al-Salam-Carlitz polynomials. AMS Contemporary Mathematics, 254:45-53 (2000)
[4] Andrews, G.E. Two theorems of Gauss and allied identities proved arithmetically. Pacific J.
Math. 41:563-578 (1972)
[5] Andrews, G.E. The Theory of Partitions, The Encyclopedia of Mathematics and Its Applications Series, Addison-Wesley Pub. Co., NY, 300 pp. (1976). Reissued, Cambridge University
Press, New York, 1998.
[6] Bessenrodt, C. A bijection for Lebesgue’s partition identity in the spirit of Sylvester. Discrete
Math. 132 (13) 1-10 (1994)
[7] Garrett, Kristina C.: A determinant identity that implies Rogers-Ramanujan. Electronic Journal
of Combinatorics 12 (2005).
[8] Keith,
William
J.
Ranks
of
Partitions
and
Durfee
Symbols.
Ph.D.
Thesis,
Pennsylvania
State
University,
2007.
Published
online
at
http://etda.libraries.psu.edu/theses/approved/WorldWideIndex/ETD-2026/index.html
[9] Pak, Igor. Partition bijections, a survey. The Ramanujan Journal 12 (1), August 2006, pp. 5-75.
URL: http://dx.doi.org/10.1007/s11139-006-9576-1
[10] Stockhofe, D.: Bijektive Abbildungen auf der Menge der Partitionen einer naturlichen Zahl.
Bayreuth. Math. Schr. (10), 1-59 (1982)
[11] Yee, Ae Ja: A combinatorial proof of Andrews’ partition functions related to Schur’s partition
theorem, Proceedings of the Amer. Math. Soc., 130 (2002), 2229-2235
[12] Želobenko, D.P. Compact Lie Groups and their Representations. Translations of Mathematical
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[13] Zeng, J.: The q-variations of Sylvesters bijection between odd and strict Partitions.” Ramanujan Journal 9(3), 289303 (2005)
William J. Keith
Drexel University
3141 Chestnut
Philadelphia, PA 19104
USA
e-mail: wjk26@drexel.edu