Polynomial Sequences of
Binomial Type
Richard P. Stanley
M.I.T.
Polynomial Sequences of Binomial Type – p.
Some motivation
Let D =
d
dn ,
acting on f (n) ∈ C[n]. Then
Dnk = knk−1
k
X
n
f (n) =
Dk f (0) (Taylor series).
k!
k≥0
Polynomial Sequences of Binomial Type – p.
Some motivation
Let D =
d
dn ,
acting on f (n) ∈ C[n]. Then
Dnk = knk−1
k
X
n
f (n) =
Dk f (0) (Taylor series).
k!
k≥0
Let ∆f (n) = f (n + 1) − f (n) and
(n)k = n(n − 1) · · · (n − k + 1). Then
∆(n)k = k(n)k−1
X
(n)k
k
f (n) =
.
∆ f (0)
k!
k≥0
Polynomial Sequences of Binomial Type – p.
Connection between D and ∆
By Taylor’s theorem,
f (n + x) =
X
k≥0
k
x
Dk f (n) .
k!
Polynomial Sequences of Binomial Type – p.
Connection between D and ∆
By Taylor’s theorem,
f (n + x) =
X
k≥0
k
x
Dk f (n) .
k!
Put x = 1:
Polynomial Sequences of Binomial Type – p.
Connection (continued)
f (n + 1) =
X Dk
k≥0
k!
!
f (n)
= eD f (n).
Polynomial Sequences of Binomial Type – p.
Connection (continued)
f (n + 1) =
X Dk
k≥0
k!
!
f (n)
= eD f (n).
⇒ ∆f (n) = (eD − 1)f (n) ⇒ ∆ = eD − 1.
Thus also D = log(∆ + 1).
Polynomial Sequences of Binomial Type – p.
Finite operator calculus
General theory developed by G.-C. Rota and
collaborators, called finite operator calculus.
Polynomial Sequences of Binomial Type – p.
Finite operator calculus
General theory developed by G.-C. Rota and
collaborators, called finite operator calculus.
G.-C. Rota, Finite Operator Calculus, Academic
Press, 1976.
Polynomial Sequences of Binomial Type – p.
The shift operator
Define E : C[n] → C[n] by
Ef (n) = f (n + 1).
Polynomial Sequences of Binomial Type – p.
Main thm. of operator calculus
Theorem. Let L : C[n] → C[n] be linear (over C)
and satisfy L(n) = 1 and L(deg d) = deg d − 1.
The following two conditions are equivalent.
LE = EL
Polynomial Sequences of Binomial Type – p.
Main thm. of operator calculus
Theorem. Let L : C[n] → C[n] be linear (over C)
and satisfy L(n) = 1 and L(deg d) = deg d − 1.
The following two conditions are equivalent.
LE = EL
There exist polynomials pk (n), k ≥ 0, such
that p0 (n) = 1, deg pk (n) = k, and
Lpk (n) = kpk−1 (n)
!n
X
X
xk
xk
=
.
pk (1)
pk (n)
k!
k!
k≥0
k≥0
Polynomial Sequences of Binomial Type – p.
Binomial type
If p0 (n), p1 (n), . . . is a sequence of polynomials
satisfying
!n
X
X
xk
xk
,
pk (n) =
pk (1)
k!
k!
k≥0
k≥0
then we call p0 (n), p1 (n), . . . a sequence of
polynomials of binomial type, or just
polynomials of binomial type.
Polynomial Sequences of Binomial Type – p.
Further properties
If LE = EL and Lpk (n) = kpk−1 (n), then:
X
pk (n)
k
f (n) =
L f (0)
n!
k≤0
(Taylor series analogue)
L is a power series in D.
...
Polynomial Sequences of Binomial Type – p.
A characterization
Note. The condition deg pk (n) = k is then
equivalent to p1 (n) 6= 0 (or just p1 (1) 6= 0).
Sometimes this extra condition is part of the
definition of binomial type.
Polynomial Sequences of Binomial Type – p. 1
A characterization
Note. The condition deg pk (n) = k is then
equivalent to p1 (n) 6= 0 (or just p1 (1) 6= 0).
Sometimes this extra condition is part of the
definition of binomial type.
Theorem. A sequence p0 (n) = 1, p1 (n), . . . of
polynomials is of binomial type if and only if
pk (m + n) =
k X
k
i=0
i
pi (m)pk−i (n), k ≥ 0.
Polynomial Sequences of Binomial Type – p. 1
Some classical examples
pk (n) = nk
Polynomial Sequences of Binomial Type – p. 1
Some classical examples
pk (n) = nk
X
k≥0
k
x
nk =
k!
X xk
k≥0
k!
!n
= enx
Polynomial Sequences of Binomial Type – p. 1
Some classical examples
pk (n) = nk
X
k≥0
k
x
nk =
k!
X xk
k≥0
k!
!n
= enx
pk (n) = (n)k = n(n − 1) · · · (n − k + 1)
Polynomial Sequences of Binomial Type – p. 1
Some classical examples
pk (n) = nk
X
k≥0
k
x
nk =
k!
X xk
k≥0
k!
!n
= enx
pk (n) = (n)k = n(n − 1) · · · (n − k + 1)
X
k≥0
k
x
(n)k =
k!
X n
k≥0
k
k
n
x = (1 + x)
Polynomial Sequences of Binomial Type – p. 1
More classical examples
pk (n) = n(k) = n(n + 1) · · · (n + k − 1)
Polynomial Sequences of Binomial Type – p. 1
More classical examples
pk (n) = n(k) = n(n + 1) · · · (n + k − 1)
X
k≥0
k
x
n(k) = (1 − x)−n
k!
Polynomial Sequences of Binomial Type – p. 1
More classical examples
pk (n) = n(k) = n(n + 1) · · · (n + k − 1)
X
k≥0
k
x
n(k) = (1 − x)−n
k!
pk (n) = n(n − ak)k−1 , a ∈ C
(Abel polynomials)
Polynomial Sequences of Binomial Type – p. 1
More classical examples
pk (n) = n(k) = n(n + 1) · · · (n + k − 1)
X
k≥0
k
x
n(k) = (1 − x)−n
k!
pk (n) = n(n − ak)k−1 , a ∈ C
(Abel polynomials)
X
k≥0
k
x
n(n − ak)k−1 =
k!
X
k≥0
k
x
(1 − ak)k−1
k!
!n
Polynomial Sequences of Binomial Type – p. 1
More on Abel polynomials
Binomial type is equivalent to Abel’s identity:
(x + y)k =
k
X
i=0
k
x(x − iz)i−1 (y + iz)k−i .
i
Note that z = 0 gives the binomial theorem.
Polynomial Sequences of Binomial Type – p. 1
More on Abel polynomials
Binomial type is equivalent to Abel’s identity:
(x + y)k =
k
X
i=0
k
x(x − iz)i−1 (y + iz)k−i .
i
Note that z = 0 gives the binomial theorem.
Closely related to tree enumeration.
Polynomial Sequences of Binomial Type – p. 1
Yet another example
pk (n) =
Pk
i=1
S(k, i)
| {z }
ni
Stirling no. of 2nd kind
(exponential polynomials)
Polynomial Sequences of Binomial Type – p. 1
Yet another example
pk (n) =
Pk
i=1
S(k, i)
| {z }
ni
Stirling no. of 2nd kind
(exponential polynomials)
X P
k≥0

k
X
i x

i S(k, i)n k! = 
k≥0
B(k)
| {z }
n
Bell number
xk 

k! 
Polynomial Sequences of Binomial Type – p. 1
One more
pk (n) =
k
X
i=1
k k−i i
i n
i
Polynomial Sequences of Binomial Type – p. 1
One more
pk (n) =
k
X
i=1
k k−i i
i n
i
!
k
X X k
x
ik−i ni
= exp nxex
k!
i
i
k≥0
Polynomial Sequences of Binomial Type – p. 1
More examples?
Are there interesting examples of polynomials of
binomial type for which explicit formulas don’t
exist?
Polynomial Sequences of Binomial Type – p. 1
Binomial posets
P = P0 ∪ P1 ∪ · · · (disjoint union): a poset
(partially ordered set) such that all maximal
chains have the form t0 < t1 < . . . , where ti ∈ Pi .
Polynomial Sequences of Binomial Type – p. 1
Binomial posets
P = P0 ∪ P1 ∪ · · · (disjoint union): a poset
(partially ordered set) such that all maximal
chains have the form t0 < t1 < . . . , where ti ∈ Pi .
Write rank(ti ) = i. Then P is a binomial poset if
for all s ≤ t, where k = rank(t) − rank(s), the
number of (saturated) chains
s = t0 < t1 < · · · < tk = t depends only on k.
Call this number B(k) (factorial function of P ).
Polynomial Sequences of Binomial Type – p. 1
Chains
P = {0, 1, 2, . . . } (a chain): B(k) = 1.
..
.
}
n=2
Polynomial Sequences of Binomial Type – p. 1
Two further examples
P = B, the set of all finite subsets of
{1, 2, . . . }, ordered by inclusion: B(k) = k!.
Polynomial Sequences of Binomial Type – p. 1
Two further examples
P = B, the set of all finite subsets of
{1, 2, . . . }, ordered by inclusion: B(k) = k!.
P = B(q), the set of all finite-dimensional
subspaces of an infinite-dimensional vector
space over the finite field Fq :
B(k) = (k)! = (1+q)(1+q+q 2 ) · · · (1+q+· · ·+q k−1 ).
Polynomial Sequences of Binomial Type – p. 1
Two further examples
P = B, the set of all finite subsets of
{1, 2, . . . }, ordered by inclusion: B(k) = k!.
P = B(q), the set of all finite-dimensional
subspaces of an infinite-dimensional vector
space over the finite field Fq :
B(k) = (k)! = (1+q)(1+q+q 2 ) · · · (1+q+· · ·+q k−1 ).
B(q) is a q-analogue of B.
Polynomial Sequences of Binomial Type – p. 1
Multichains
Theorem. Let P be a binomial poset. Let pk (n)
be the number of multichains
s = t0 ≤ t1 ≤ t2 ≤ · · · ≤ tn = t,
where rank(t) − rank(s) = k. Then
X
k≥0
xk
pk (n)
=
B(k)
X xk
B(k)
k≥0
!n
.
Polynomial Sequences of Binomial Type – p. 2
Multichains
Theorem. Let P be a binomial poset. Let pk (n)
be the number of multichains
s = t0 ≤ t1 ≤ t2 ≤ · · · ≤ tn = t,
where rank(t) − rank(s) = k. Then
X
k≥0
xk
pk (n)
=
B(k)
X xk
B(k)
k≥0
!n
.
Corollary. k! pk (n)/B(k), k ≥ 0, is a sequence of
polynomials of binomial type.
Polynomial Sequences of Binomial Type – p. 2
Example: B(q)
Let rk (n) be the number of multichains of
subspaces
{0} = V0 ⊆ V1 ⊆ · · · ⊆ Vn = Fkq .
Let
k! rk (n)
pk (n) =
.
(k)!
Then p0 (n), p1 (n), . . . is a sequence of
polynomials of binomial type.
Polynomial Sequences of Binomial Type – p. 2
Other example
Many other examples of binomial posets, e.g.,
1
B(k) = k .
2(2) k!
Related to graph colorings and acyclic
orientations.
Polynomial Sequences of Binomial Type – p. 2
New vistas: toroidal graphs
Zdn : the n × n × · · · × n (d times) d-dimensional
toroidal graph.
Polynomial Sequences of Binomial Type – p. 2
New vistas: toroidal graphs
Zdn : the n × n × · · · × n (d times) d-dimensional
toroidal graph.
Z24
Polynomial Sequences of Binomial Type – p. 2
New vistas: toroidal graphs
Zdn : the n × n × · · · × n (d times) d-dimensional
toroidal graph.
The green squares are the vertices of Z24 .
Polynomial Sequences of Binomial Type – p. 2
Algebraic definition
Zn : integers modulo n
Zdn : {(a1 , . . . , ad ) : ai ∈ Zn } (vertex set)
α = (a1 , . . . , ad ) and β = (b1 , . . . , bd ) are adjacent
if α − β has one nonzero coordinate, which is
equal to ±1 (modulo n).
Polynomial Sequences of Binomial Type – p. 2
Figures
{
a set S of figures:
}
Polynomial Sequences of Binomial Type – p. 2
Figures
{
a set S of figures:
}
A placement of S on Z24 :
Polynomial Sequences of Binomial Type – p. 2
d
The function fk (n )
Fix d and a finite set S of tiles.
fk (nd ): number of placements of S on Zdn
covering a total of k 1 × 1 × · · · × 1 boxes.
Polynomial Sequences of Binomial Type – p. 2
d
The function fk (n )
Fix d and a finite set S of tiles.
fk (nd ): number of placements of S on Zdn
covering a total of k 1 × 1 × · · · × 1 boxes.
n2
2
Example. S = {}. Then fk (n ) = k
Polynomial Sequences of Binomial Type – p. 2
Another example
Example. S =
{
}
f2j+1 (n2 ) = 0
f2 (n2 ) = n2
1 2 2
2
f4 (n ) = n (n − 3)
2
Polynomial Sequences of Binomial Type – p. 2
Still another example
Example. S = {
}
f1 (n2 ) = n2
2
n
2
f2 (n ) =
+ n2
2
2
n
2
+ n2 (n2 − 2)
f3 (n ) =
3
Polynomial Sequences of Binomial Type – p. 2
Still another example
Example. S = {
}
f1 (n2 ) = n2
2
n
2
f2 (n ) =
+ n2
2
2
n
2
+ n2 (n2 − 2)
f3 (n ) =
3
Note that these are polynomials in n2 .
Polynomial Sequences of Binomial Type – p. 2
Relationship to binomial type
Theorem (Jon Schneider)
(a) For n ≫ 0 (so all tiles fit on Zdn ), there is a
polynomial pk for which pk (n) = fk (nd ).
Polynomial Sequences of Binomial Type – p. 2
Relationship to binomial type
Theorem (Jon Schneider)
(a) For n ≫ 0 (so all tiles fit on Zdn ), there is a
polynomial pk for which pk (n) = fk (nd ).
(b) p0 , 1! p1 , 2! p2 , . . . is a sequence of polynomials
of binomial type.
Polynomial Sequences of Binomial Type – p. 2
An example
Recall: S = {
}
2
n
2
2
2
+ n2
f1 (n ) = n , f2 (n ) =
2
2
n
2
f3 (n ) =
+ n2 (n2 − 2)
3
Polynomial Sequences of Binomial Type – p. 3
An example
Recall: S = {
}
2
n
2
2
2
+ n2
f1 (n ) = n , f2 (n ) =
2
2
n
2
f3 (n ) =
+ n2 (n2 − 2)
3
n
n
2
1 + nx +
+n x +
+ n(n − 2) x3
2
3
+ · · · = (1 + x + x2 − x3 + · · · )n
Polynomial Sequences of Binomial Type – p. 3
Chromatic polynomials
G: finite graph with vertex set V , q ≥ 1
χG (q): number of proper colorings
f : V → {1, . . . , q},
i.e., adjacent vertices get different colors
Polynomial Sequences of Binomial Type – p. 3
Chromatic polynomials
G: finite graph with vertex set V , q ≥ 1
χG (q): number of proper colorings
f : V → {1, . . . , q},
i.e., adjacent vertices get different colors
Example. G = Kn , complete graph with n
vertices. Then
χKn (q) = q(q − 1) · · · (q − n + 1).
Polynomial Sequences of Binomial Type – p. 3
The graph
d
Zn
Recall Zdn is a graph:
Z24
Much interest from physicists in the chromatic
polynomial χZdn (q).
Polynomial Sequences of Binomial Type – p. 3
A trivial and nontrivial result
χZdn (2) =
(
2, n even
0, n odd
Polynomial Sequences of Binomial Type – p. 3
A trivial and nontrivial result
χZdn (2) =
(
2, n even
0, n odd
Theorem (E. Lieb, 1967)
3/2
4
1/n2
lim χZ2n (3)
=
= 1.5396 · · ·
n→∞
3
(residual entropy of square ice)
Polynomial Sequences of Binomial Type – p. 3
Open variants
1/n2
lim χZ2n (3)
n→∞
3/2
4
=
3
1/n2
: not known
1/n3
: not known
limn→∞ χZ2n (4)
limn→∞ χZ3n (3)
Polynomial Sequences of Binomial Type – p. 3
Broken circuits
Label the edges of the graph G as 1, 2, . . . , m.
broken circuit: a circuit with its largest edge
removed
3
3
8
7
4
7
4
2
2
circuit
broken circuit
Polynomial Sequences of Binomial Type – p. 3
The broken circuit theorem
Theorem (H. Whitney, 1932) Let G have N
vertices. Write
χG (q) = a0 q N − a1 q N −1 + a2 q N −2 − · · · .
Then ai is the number of i-element sets of edges
of G that contain no broken circuit.
Polynomial Sequences of Binomial Type – p. 3
An example
Example. If G is a 4-cycle, then no 0-element,
1-element, or 2-element set of edges contains a
broken circuit. One 3-element set contains (in
fact, is) a broken circuit, and all four edges
contain a broken circuit. Hence
4 3
4 2
4
4
χG (q) = q −
q +
q −
−1 q
1
2
3
= q 4 − 4q 3 + 6q 2 − 3.
Polynomial Sequences of Binomial Type – p. 3
Broken circuits in
2
Zn
Let G = Z2n and N = n2 (number of vertices), so
2N edges. The smallest cycle in G has length
four. There are N such cycles, so N 3-element
sets of edges containing (in fact, equal to) a
broken circuit. Hence
2N N −1
2N N −2
N
χZ2n (q) = q −
q
+
q
1
2
2N
N −3
−
−N q
+ ···
3
Polynomial Sequences of Binomial Type – p. 3
Chromatic polynomial of
d
Zn
Theorem (J. Schneider). Let N = nd , the
number of vertices of Zdn . Write
χZdn (q) = c0 (N )q N −c1 (N )q N −1 +c2 (N )q N −2 −· · · .
Then for N ≫ 0, ck (N ) agrees with a polynomial
pk (N ). Moreover, p0 , 1! p1 , 2! p2 , . . . is a sequence
of polynomials of binomial type.
Polynomial Sequences of Binomial Type – p. 3
Chromatic polynomial of
d
Zn
Theorem (J. Schneider). Let N = nd , the
number of vertices of Zdn . Write
χZdn (q) = c0 (N )q N −c1 (N )q N −1 +c2 (N )q N −2 −· · · .
Then for N ≫ 0, ck (N ) agrees with a polynomial
pk (N ). Moreover, p0 , 1! p1 , 2! p2 , . . . is a sequence
of polynomials of binomial type.
Proof uses a variant of Schneider’s previous
result on placing tiles on Zdn .
Polynomial Sequences of Binomial Type – p. 3
Computations
Let d = 2. D. Kim and I. G. Enting made a
computation (1979) equivalent to
X
pk (N )xk = (1 + 2x + x2 − x3 + x4 − x5 + x6
k≥0
−2x7 + 9x8 − 38x9 + 130x10
−378x11 + 987x12 − 2436x13
+5927x14 − 14438x15 + 34359x16
−75058x17 + 134146x18 + · · · )N .
Polynomial Sequences of Binomial Type – p. 4
Computations
Let d = 2. D. Kim and I. G. Enting made a
computation (1979) equivalent to
X
pk (N )xk = (1 + 2x + x2 − x3 + x4 − x5 + x6
k≥0
−2x7 + 9x8 − 38x9 + 130x10
−378x11 + 987x12 − 2436x13
+5927x14 − 14438x15 + 34359x16
−75058x17 + 134146x18 + · · · )N .
Can anything be said about these numbers?
Does the series converge for small x?
Polynomial Sequences of Binomial Type – p. 4
Some small values
p1 (N )
p2 (N )
p3 (N )
p4 (N )
p5 (N )
p6 (N )
=
=
=
=
=
=
2N
2N (2N − 1)
2N (4N 2 − 6N − 1)
4N (N + 1)(2N − 3)(2N − 5)
8N (N + 2)(N − 2)(2N − 3)(2N − 7)
8N (8N 5 − 60N 4 + 50N 3 + 495N 2
−1228N + 825)
p7 (N ) = 8N (16N 6 − 168N 5 + 280N 4 + 2310N 3
−10241N 2 + 14553N − 8010)
Polynomial Sequences of Binomial Type – p. 4
Further directions
What about n1 × n2 × · · · × nd tori?
Polynomial Sequences of Binomial Type – p. 4
Further directions
What about n1 × n2 × · · · × nd tori?
Nothing new: simply replace N = nd with
N = n1 n2 · · · nd .
Polynomial Sequences of Binomial Type – p. 4
Tutte polynomials
What about replacing chromatic polynomials with
Tutte polynomials?
Polynomial Sequences of Binomial Type – p. 4
Tutte polynomials
What about replacing chromatic polynomials with
Tutte polynomials?
Currently under investigation. No known
satisfactory generalization of broken circuit
theorem.
Polynomial Sequences of Binomial Type – p. 4
Multi-indexed polynomials
What about replacing pk (n) with pj,k (n)?
Polynomial Sequences of Binomial Type – p. 4
Multi-indexed polynomials
What about replacing pk (n) with pj,k (n)?
To be investigated.
Polynomial Sequences of Binomial Type – p. 4
Multi-indexed polynomials
What about replacing pk (n) with pj,k (n)?
To be investigated.
Interesting example. Kjk : complete bipartite
graph
Theorem (EC2, Exercise 5.6).
X
j,k≥0
xj y k
= (ex + ey − 1)n
χKjk (n)
j! k!
Polynomial Sequences of Binomial Type – p. 4
Multivariate polynomials
What about replacing pk (n) with pk (m, n)?
Polynomial Sequences of Binomial Type – p. 4
Multivariate polynomials
What about replacing pk (n) with pk (m, n)?
Not yet considered.
May involve F (x)m G(x)n .
Polynomial Sequences of Binomial Type – p. 4
Continuous variant
Theorem (Schneider). Let S be a bounded
measurable set in d-dimensional Euclidean
space. Let Pk (nd ) be the probability that no two
copies intersect when we place k copies of S
independently and uniformly at random inside a
d-dimensional torus of side length n. Then
ndk Pk (nd ) is eventually a polynomial pk (n) for
each k, and these polynomials form a sequence
of binomial type.
Polynomial Sequences of Binomial Type – p. 4
A reference
arXiv:1206.6174
Polynomial Sequences of Binomial Type – p. 4
The last slide
Polynomial Sequences of Binomial Type – p. 4
The last slide
Polynomial Sequences of Binomial Type – p. 4
The last slide
Polynomial Sequences of Binomial Type – p. 4