Discrete Mathematics and Theoretical Computer Science 6, 2003, 101–106
Fountains, histograms, and q–identities
Peter Paule1† and Helmut Prodinger2‡
1 Research
Institute for Symbolic Computation, Johannes Kepler University Linz, A-4040 Linz, Austria
e-mail: Peter.Paule@risc.uni-linz.ac.at
Url: http://www.risc.uni-linz.ac.at/research/combinat/
2
The John Knopfmacher Centre for Applicable Analysis and Number Theory, School of Mathematics, University of
the Witwatersrand, P. O. Wits, 2050 Johannesburg, South Africa
e-mail: helmut@maths.wits.ac.za
Url: http://www.wits.ac.za/helmut/
received Jan 14, 2003, accepted Aug 15, 2003.
We solve the recursion Sn = Sn−1 − qn Sn−p , both, explicitly, and in the limit for n → ∞, proving in this way a formula
due to Merlini and Sprugnoli. It is also discussed how computer algebra could be applied.
Keywords: q–identities, fountains, histograms, Schur polynomials
1
Fountains and histograms
Merlini and Sprugnoli [6] discuss fountains and histograms; for the reader’s convenience, we review a
few key issues here.
A fountain with n coins is an arrangement of n coins in rows such that each coin in a higher row touches
exactly two coins in the next lower row.
A p-histogram is a sequence of columns in which the height of the ( j + 1)st column is at most k + p, if
k is the height of column j; the first column has height r, with 1 ≤ r ≤ p.
It can be shown that the enumeration of coins in a fountain is equivalent with the enumeration of
1-histograms. The paper [6] addresses the enumeration of p-histograms with respect to area (=number of
[p]
cells). Let fn be the number p-histograms with area n and F [p] (q) the corresponding generating function
[p]
F [p] (q) = ∑n fn qn . The authors of [6] use two different approaches: one produces the answer in the form
F [p] (q) = lim
m→∞
Dm
,
Em
† Partially
‡ The
supported by SFB grant F1305 of the Austrian FWF.
research was started when this author visited the RISC in 2002. Partially supported by NRF grant 2053748.
c 2003 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France
1365–8050 102
Peter Paule and Helmut Prodinger
with some polynomials Dm , Em defined in the next section, and the other gives it as
k+1
k
(−1)k q p( 2 )
(−1)k qk+p(2)
F (q) = ∑
.
∑
k
k
k≥0 (1 − q) . . . (1 − q )
k≥0 (1 − q) . . . (1 − q )
[p]
According to [5], it would be nice to have a direct argument that these two answers coincide. This is
the subject of the present note.
2
Generalized Schur polynomials
The polynomials mentioned in the introduction are for fixed p ≥ 1 defined as follows:
En = En−1 − qn En−p ,
n ≥ p,
E0 = · · · = E p−1 = 1,
Dn = Dn−1 − qn Dn−p ,
n ≥ p,
Di = 1 − ∑ q j , i = 0, . . . , p − 1.
i
j=1
They can be compared with the classical Schur polynomials [8], which occur for p = 2 and q = −1. Then
Merlini and Sprugnoli want a direct proof of the formulæ
E∞ := lim En =
n→∞
k+1
(−1)k q p( 2 )
,
∑
k
k≥0 (1 − q) . . . (1 − q )
k
(−1)k qk+p(2)
D∞ := lim Dn = ∑
.
k
n→∞
k≥0 (1 − q) . . . (1 − q )
We will not only achieve that but actually derive explicit expressions for these polynomials!
It should be mentioned that Cigler [4] developed independently a combinatorial method to deal with
recursions as ours, but also more general ones.
Let us study the generic recursion
Sn = Sn−1 + tqn−p Sn−p ,
with unspecified initial values S0 , . . . , S p−1 . For p = 2, these polynomials were studied by Andrews (and
others) in the context of Schur polynomials, see [2].
We will use standard notation from q–calculus, see [1]:
n
(q)n
(x)n = (1 − x)(1 − xq) . . . (1 − xqn−1 ),
=
.
(q)k (q)n−k
k
It will be convenient to define nk = 0 for n < 0 or k > n.
Now we will proceed as in [1] and consider noncommutative variables x, η, such that xη = qηx; all
other variables commute.
Lemma 1.
n n
n p(n)−pnk+p(k+1) k+p(n−k) n−k
p
2 x
q 2
x+x η = ∑
η .
k=0 k
Fountains, histograms, and q–identities
103
Proof. We write
n
x + xpη =
n
∑ an,k xk+p(n−k) ηn−k ,
k=0
n+1
n
n+1
n
and x + x p η
= x + x p η x + x p η resp. as x + x p η
= x + x p η x + x p η , compare coefficients, and get the recursions
an+1,k = an,k−1 + an,k qk+p(n−k) ,
an+1,k = an,k−1 qn+1−k + an,k q p(n−k) .
From this we derive, taking differences,
an,k =
1 − qn+1−k −p(n−k)
q
an,k−1 .
1 − qk
n
The result follows from iteration by noting that an,0 = q p(2) .
Of course we also have
n n
n p(n)−pnk+p(k+1) k+p(n−k) n−k n−k
2 x
t η .
x + tx p η = ∑
q 2
k=0 k
Now we derive the generating function for
F(x) =
∑ Sn xn ;
n≥0
the following procedure is inspired by [2]. Note that we can alternatively view η as an operator, defined
by η f (x) = f (qx). Cigler worked also much with this technique [3, 4]. We find
∑ Sn xn = ∑ Sn−1 xn + ∑ tqn−p Sn−p xn = x ∑
n≥p
n≥p
n≥p
or
F(x) −
n≥p−1
∑ Sn xn = xF(x) − x ∑
n<p
1
F(x) =
1 − x − tx p η
Now we can apply our lemma and write
n
p
F(x) = ∑ x + tx η
∑ Si x i −
i<p
n
∑ ηSn xn
n≥0
Sn xn + tx p ηF(x),
n<p−1
and
n≥0
Sn xn + tx p
∑
Si x
∑ Si x − ∑
i
i<p
i+1
Si x
i+1
.
i<p−1
i<p−1
n p(n)−pnk+p(k+1) k+p(n−k) n−k n−k
i
i+1
2
2
=∑∑
q
x
t η
∑ Si x − ∑ Si x
i<p
n≥0 k=0 k
i<p−1
n n p(n)−pnk+p(k+1) k+p(n−k) n−k
i(n−k) i
(i+1)(n−k) i+1
2
2
q
=∑∑
x
t
x
∑ Si q x − ∑ Si q
i<p
n≥0 k=0 k
i<p−1
104
Peter Paule and Helmut Prodinger
n
n p(k) n−k+pk k
2 x
q
t
∑∑ k
∑ Si qik xi − ∑ Si q(i+1)k xi+1
i<p
i<p−1
n≥0 k=0
n + k p(k) n+pk k
(i+1)k
i+1
ik
i
x
q 2 x
t ∑ Si q x − ∑ Si q
= ∑
k
i<p
i<p−1
k,n≥0
1
p(2k) pk k
(i+1)k i+1
ik i
= ∑q
x t
∑ Si q x − ∑ Si q x .
(x)k+1 i<p
i<p−1
k≥0
=
From this we find an explicit formula for Sn (the quantity S−1 has to be interpreted as 0):
Sn =
∑
n − (p − 1)k − i p(k)+ik k
q 2
t .
k
k≥0
(Si − Si−1 ) ∑
0≤i<p
Now we specialize this to our instance. Here, t = −q p , and thus
n − (p − 1)k − i p(k+1)+ik
Sn = ∑ (Si − Si−1 ) ∑
(−1)k .
q 2
k
0≤i<p
k≥0
Therefore
En =
n − (p − 1)k p(k+1)
q 2 (−1)k .
∑
k
k≥0
From this, the limit of En is immediate. For Dn we eventually get the following form
n − (p − 1)(k − 1) k+p(k)
2 (−1)k ,
Dn = ∑
q
k
k≥0
from which the formula for D∞ is immediate. To prove it, we need a simple lemma whose proof is just a
routine calculation.
Lemma 2.
m − i i(k+1)
m − i (i+1)(k+1)
q
= g(i) − g(i − 1)
where
g(i) = −
q
.
k
k+1
Now we can plug into the general formula above and compute
p−1
n − (p − 1)k − i p(k+1)+i(k+1)
Dn = En − ∑ ∑
q 2
(−1)k
k
i=1 k≥0
p−1 k+1
n − (p − 1)k − i i(k+1)
= En − ∑ (−1)k q p( 2 ) ∑
q
k
i=1
k≥0
k+1
n − (p − 1)k
n − (p − 1)(k + 1)
− q p(k+1)
= En − ∑ (−1)k q p( 2 ) qk+1
k+1
k+1
k≥0
k+1
n − (p − 1)k
= 1 − ∑ (−1)k q p( 2 ) qk+1
,
k+1
k≥0
which is the announced formula after a simple change of variable. Note that in the penultimate step the
telescoping property of the lemma has been used.
Fountains, histograms, and q–identities
3
105
Computer algebra proofs
The polynomial families (En ) and (Dn ) give rise to the following study with respect to possible computer
proofs. Let us take as input our sum representations of En and Dn :
n − (p − 1)k p(k+1)
q 2 (−1)k ,
∑
k
k≥0
n − (p − 1)(k − 1) k+p(k)
2 (−1)k .
Dn = ∑
q
k
k≥0
En =
(3.1)
Then, if p is chosen as a specific positive integer, Riese’s package qZeil [7] returns the recurrences Sn =
Sn−1 − qn Sn−p (n ≥ p) together with a certificate function Cert for independent verification. Despite the
fact that for general “generic” integer parameter p there is no algorithm available, a general pattern can
be easily guessed from running the algorithm for p = 1, p = 2, and p = 3, say.
For example, let F(n, k) be the kth summand in our sum representation (3.1) of En , then the recurrence
for En can be refined to the following statement.
Theorem 3.1. For n ≥ p and δk f (n, k) = f (n, k) − f (n, k − 1), we have
F(n, k) − F(n − 1, k) + qn F(n − p, k) = δk Cert(n, k)F(n, k),
(3.2)
where
Cert(n, k) = qn
(qn−p(k+1)+1 ) p
.
(qn−(p−1)(k+1) ) p
Proof. After dividing both sides of (3.2) by F(n, k) the proof reduces to checking equality of rational
functions. Namely, note that
1 − qn−pk
F(n − 1, k)
=
,
F(n, k)
1 − qn−(p−1)k
(qn−pk+1 ) p
F(n, k − 1)
q pk
=−
,
F(n, k)
1 − qk (qn−(p−1)k+1 ) p−1
and
F(n − p, k)
= q−n Cert(n, k).
F(n, k)
Analogously, there is a refined version of the recurrence for Dn . The certificate in this case is
Cert(n, k) = qn
(qn−pk ) p
.
(qn−(p−1)k ) p
Summarizing, with the sum representation for En and Dn in hand, the corresponding recurrences follow
immediately by summing both sides of the computer recurrences (3.2) over all k ≥ 0.
106
Peter Paule and Helmut Prodinger
References
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[2] G. E. Andrews, Fibonacci numbers and Rogers–Ramanujan identities, The Fibonacci Quarterly, to
appear (2003), 15 pp.
[3] J. Cigler, Elementare q–Identitäten, Séminaire Lotharingien de Combinatoire B05a (1981), 29 pp.
[4] J. Cigler, Some algebraic aspects of Morse code sequences, Discrete Mathematics and Theoretical
Computer Science 6 (2003), 55–68.
[5] D. Merlini, Private communication, (2002).
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pp. 179–210.
[8] I. Schur, Ein Beitrag zur additiven Zahlentheorie und zur Theorie der Kettenbrüche, S.-B. Preuss.
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