Polynomial Sequences of Binomial Type

Richard P. Stanley

M.I.T.

Polynomial Sequences of Binomial Type – p. Some motivation

d C Let D = dn, acting on f(n) ∈ [n]. Then Dnk = knk−1 nk f(n) = Dkf(0) (Taylor series). k! Xk≥0

Polynomial Sequences of Binomial Type – p. Some motivation

d C Let D = dn, acting on f(n) ∈ [n]. Then Dnk = knk−1 nk f(n) = Dkf(0) (Taylor series). k! Xk≥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 (n) f(n) = ∆kf(0) k . k! k≥0 X Polynomial Sequences of Binomial Type – p. Connection between D and ∆

By Taylor’s theorem,

xk f(n + x)= Dkf(n) . k! Xk≥0

Polynomial Sequences of Binomial Type – p. Connection between D and ∆

By Taylor’s theorem,

xk f(n + x)= Dkf(n) . k! Xk≥0 Put x = 1:

Polynomial Sequences of Binomial Type – p. Connection (continued)

Dk f(n +1) = f(n) k! ! Xk≥0 = eDf(n).

Polynomial Sequences of Binomial Type – p. Connection (continued)

Dk f(n +1) = f(n) k! ! Xk≥0 = eDf(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 xk xk pk(n) = pk(1) . k! k! ! Xk≥0 Xk≥0

Polynomial Sequences of Binomial Type – p. Binomial type

If p0(n),p1(n),... is a sequence of polynomials satisfying

n xk xk pk(n) = pk(1) , k! k! ! Xk≥0 Xk≥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: p (n) f(n)= Lkf(0) k n! k≤0 (Taylor seriesX 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. 10 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

k k p (m + n)= p (m)p (n), k ≥ 0. k i i k−i i=0 X  

Polynomial Sequences of Binomial Type – p. 10 Some classical examples

k pk(n)= n

Polynomial Sequences of Binomial Type – p. 11 Some classical examples

k pk(n)= n

n xk xk nk = = enx k! k! ! Xk≥0 Xk≥0

Polynomial Sequences of Binomial Type – p. 11 Some classical examples

k pk(n)= n

n xk xk nk = = enx k! k! ! Xk≥0 Xk≥0

pk(n) = (n)k = n(n − 1) ··· (n − k + 1)

Polynomial Sequences of Binomial Type – p. 11 Some classical examples

k pk(n)= n

n xk xk nk = = enx k! k! ! Xk≥0 Xk≥0

pk(n) = (n)k = n(n − 1) ··· (n − k + 1)

xk n (n) = xk = (1+ x)n k k! k Xk≥0 Xk≥0  

Polynomial Sequences of Binomial Type – p. 11 More classical examples

(k) pk(n)= n = n(n + 1) ··· (n + k − 1)

Polynomial Sequences of Binomial Type – p. 12 More classical examples

(k) pk(n)= n = n(n + 1) ··· (n + k − 1)

xk n(k) = (1 − x)−n k! Xk≥0

Polynomial Sequences of Binomial Type – p. 12 More classical examples

(k) pk(n)= n = n(n + 1) ··· (n + k − 1)

xk n(k) = (1 − x)−n k! Xk≥0 k−1 pk(n)= n(n − ak) , a ∈ C (Abel polynomials)

Polynomial Sequences of Binomial Type – p. 12 More classical examples

(k) pk(n)= n = n(n + 1) ··· (n + k − 1)

xk n(k) = (1 − x)−n k! Xk≥0 k−1 pk(n)= n(n − ak) , a ∈ C (Abel polynomials)

n xk xk n(n − ak)k−1 = (1 − ak)k−1 k! k! ! Xk≥0 Xk≥0

Polynomial Sequences of Binomial Type – p. 12 More on Abel polynomials

Binomial type is equivalent to Abel’s identity:

k k (x + y)k = x(x − iz)i−1(y + iz)k−i. i i=0 X   Note that z = 0 gives the binomial theorem.

Polynomial Sequences of Binomial Type – p. 13 More on Abel polynomials

Binomial type is equivalent to Abel’s identity:

k k (x + y)k = x(x − iz)i−1(y + iz)k−i. i i=0 X   Note that z = 0 gives the binomial theorem. Closely related to tree enumeration.

Polynomial Sequences of Binomial Type – p. 13 Yet another example

k i pk(n)= i=1 S(k, i) n . P Stirling no of 2nd kind (exponential| {z polynomials} )

Polynomial Sequences of Binomial Type – p. 14 Yet another example

k i pk(n)= i=1 S(k, i) n . P Stirling no of 2nd kind (exponential| {z polynomials} )

n i xk xk S(k, i)n =  B(k)  i k! k! k≥0 k≥0 X   X   Bell number  P   | {z }

Polynomial Sequences of Binomial Type – p. 14 One more

k k p (n)= ik−ini k i i=1 X  

Polynomial Sequences of Binomial Type – p. 15 One more

k k p (n)= ik−ini k i i=1 X  

k k − x ik ini = exp nxex i k! i ! Xk≥0 X  

Polynomial Sequences of Binomial Type – p. 15 More examples?

Are there interesting examples of polynomials of binomial type for which explicit formulas don’t exist?

Polynomial Sequences of Binomial Type – p. 16 Binomial posets

P = P0 ∪ P1 ∪··· (disjoint union): a poset (partially ordered set) such that all maximal chains have the form t0

Polynomial Sequences of Binomial Type – p. 17 Binomial posets

P = P0 ∪ P1 ∪··· (disjoint union): a poset (partially ordered set) such that all maximal chains have the form t0

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

Polynomial Sequences of Binomial Type – p. 17 Chains

P = {0, 1, 2,... } (a chain): B(k) = 1. . .

}n = 2

Polynomial Sequences of Binomial Type – p. 18 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. 19 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+q2) ··· (1+q+···+qk−1).

Polynomial Sequences of Binomial Type – p. 19 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+q2) ··· (1+q+···+qk−1).

B(q) is a q-analogue of B.

Polynomial Sequences of Binomial Type – p. 19 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

n xk xk pk(n) = . B(k) B(k)! Xk≥0 Xk≥0

Polynomial Sequences of Binomial Type – p. 20 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

n xk xk pk(n) = . B(k) B(k)! Xk≥0 Xk≥0 Corollary. k! pk(n)/B(k), k ≥ 0, is a sequence of polynomials of binomial type.

Polynomial Sequences of Binomial Type – p. 20 Example: B(q)

Let rk(n) be the number of multichains of subspaces Fk {0} = V0 ⊆ V1 ⊆···⊆ Vn = q . Let k! r (n) p (n)= k . k (k)!

Then p0(n),p1(n),... is a sequence of polynomials of binomial type.

Polynomial Sequences of Binomial Type – p. 21 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. 22 New vistas: toroidal graphs

Zd n: the n × n ×···× n (d times) d-dimensional toroidal graph.

Polynomial Sequences of Binomial Type – p. 23 New vistas: toroidal graphs

Zd n: the n × n ×···× n (d times) d-dimensional toroidal graph.

Z2 4

Polynomial Sequences of Binomial Type – p. 23 New vistas: toroidal graphs

Zd n: the n × n ×···× n (d times) d-dimensional toroidal graph.

Z2 The green squares are the vertices of 4.

Polynomial Sequences of Binomial Type – p. 23 Algebraic definition

Zn: integers modulo n Zd Z n: {(a1,...,ad) : ai ∈ n} (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. 24 Figures

a set S of figures: {}

Polynomial Sequences of Binomial Type – p. 25 Figures

a set S of figures: {}

Z2 A placement of S on 4:

Polynomial Sequences of Binomial Type – p. 25 d The function fk(n )

Fix d and a finite set S of tiles. d Zd fk(n ): number of placements of S on n covering a total of k 1 × 1 ×···× 1 boxes.

Polynomial Sequences of Binomial Type – p. 26 d The function fk(n )

Fix d and a finite set S of tiles. d Zd fk(n ): number of placements of S on n covering a total of k 1 × 1 ×···× 1 boxes.

2  2 n Example. S = { }. Then fk(n )= k

Polynomial Sequences of Binomial Type – p. 26 Another example

Example. S = {}

2 f2j+1(n ) = 0 2 2 f2(n ) = n 1 f (n2) = n2(n2 − 3) 4 2

Polynomial Sequences of Binomial Type – p. 27 Still another example

Example. S = {}

2 2 f1(n ) = n n2 f (n2) = + n2 2 2   n2 f (n2) = + n2(n2 − 2) 3 3  

Polynomial Sequences of Binomial Type – p. 28 Still another example

Example. S = {}

2 2 f1(n ) = n n2 f (n2) = + n2 2 2   n2 f (n2) = + n2(n2 − 2) 3 3   Note that these are polynomials in n2.

Polynomial Sequences of Binomial Type – p. 28 Relationship to binomial type

Theorem (Jon Schneider) Zd (a) For n ≫ 0 (so all tiles fit on n), there is a d polynomial pk for which pk(n)= fk(n ).

Polynomial Sequences of Binomial Type – p. 29 Relationship to binomial type

Theorem (Jon Schneider) Zd (a) For n ≫ 0 (so all tiles fit on n), there is a d polynomial pk for which pk(n)= fk(n ).

(b) p0, 1! p1, 2! p2,... is a sequence of polynomials of binomial type.

Polynomial Sequences of Binomial Type – p. 29 An example

Recall: S = {} n2 f (n2)= n2, f (n2)= + n2 1 2 2   n2 f (n2)= + n2(n2 − 2) 3 3  

Polynomial Sequences of Binomial Type – p. 30 An example

Recall: S = {} n2 f (n2)= n2, f (n2)= + n2 1 2 2   n2 f (n2)= + n2(n2 − 2) 3 3   n n 1+ nx + + n x2 + + n(n − 2) x3 2 3       + ··· = (1+ x + x2 − x3 + ··· )n

Polynomial Sequences of Binomial Type – p. 30 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. 31 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. 31 Zd The graph n

Zd Recall n is a graph:

Z2 4 Much interest from physicists in the chromatic d polynomial χZn (q).

Polynomial Sequences of Binomial Type – p. 32 A trivial and nontrivial result

2, n even Zd χ n (2) = ( 0, n odd

Polynomial Sequences of Binomial Type – p. 33 A trivial and nontrivial result

2, n even Zd χ n (2) = ( 0, n odd Theorem (E. Lieb, 1967)

3/2 1/n2 4 lim χZ2 (3) = = 1.5396 ··· n→∞ n 3   (residual entropy of square ice)

Polynomial Sequences of Binomial Type – p. 33 Open variants

3/2 1/n2 4 lim χZ2 (3) = n→∞ n 3   1/n2 Z2 limn→∞ χ n (4) : not known 1/n3 Z3 limn→∞ χ n (3) : not known

Polynomial Sequences of Binomial Type – p. 34 Broken circuits

Label the edges of the graph G as 1, 2,...,m. broken circuit: a circuit with its largest edge removed

3 8 3

7 4 7 4

2 2 circuit broken circuit

Polynomial Sequences of Binomial Type – p. 35 The broken circuit theorem

Theorem (H. Whitney, 1932) Let G have N vertices. Write

N N−1 N−2 χG(q)= a0q − a1q + a2q −··· .

Then ai is the number of i-element sets of edges of G that contain no broken circuit.

Polynomial Sequences of Binomial Type – p. 36 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 4 4 χ (q) = q4 − q3 + q2 − − 1 q G 1 2 3        = q4 − 4q3 + 6q2 − 3.

Polynomial Sequences of Binomial Type – p. 37 Z2 Broken circuits in n

Z2 2 Let G = n and N = n (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

N 2N N−1 2N N−2 χZ2 (q) = q − q + q n 1 2     2N − − N qN−3 + ··· 3   

Polynomial Sequences of Binomial Type – p. 38 Zd Chromatic polynomial of n

Theorem (J. Schneider). Let N = nd, the Zd number of vertices of n. Write

N N−1 N−2 Zd c N c N c N χ n (q)= 0( )q − 1( )q + 2( )q −··· .

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. 39 Zd Chromatic polynomial of n

Theorem (J. Schneider). Let N = nd, the Zd number of vertices of n. Write

N N−1 N−2 Zd c N c N c N χ n (q)= 0( )q − 1( )q + 2( )q −··· .

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 Zd result on placing tiles on n.

Polynomial Sequences of Binomial Type – p. 39 Computations

Let d = 2. D. Kim and I. G. Enting made a computation (1979) equivalent to

k 2 3 4 5 6 pk(N)x = (1 + 2x + x − x + x − x + x 7 8 9 10 Xk≥0 −2x + 9x − 38x + 130x −378x11 + 987x12 − 2436x13 +5927x14 − 14438x15 + 34359x16 −75058x17 + 134146x18 + ··· )N .

Polynomial Sequences of Binomial Type – p. 40 Computations

Let d = 2. D. Kim and I. G. Enting made a computation (1979) equivalent to

k 2 3 4 5 6 pk(N)x = (1 + 2x + x − x + x − x + x 7 8 9 10 Xk≥0 −2x + 9x − 38x + 130x −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. 40 Some small values

p1(N) = 2N p2(N) = 2N(2N − 1) 2 p3(N) = 2N(4N − 6N − 1) p4(N) = 4N(N + 1)(2N − 3)(2N − 5) p5(N) = 8N(N + 2)(N − 2)(2N − 3)(2N − 7) 5 4 3 2 p6(N) = 8N(8N − 60N + 50N + 495N −1228N + 825) 6 5 4 3 p7(N) = 8N(16N − 168N + 280N + 2310N −10241N 2 + 14553N − 8010)

Polynomial Sequences of Binomial Type – p. 41 Further directions

What about n1 × n2 ×···× nd tori?

Polynomial Sequences of Binomial Type – p. 42 Further directions

What about n1 × n2 ×···× nd tori? Nothing new: simply replace N = nd with N = n1n2 ··· nd.

Polynomial Sequences of Binomial Type – p. 42 Tutte polynomials

What about replacing chromatic polynomials with Tutte polynomials?

Polynomial Sequences of Binomial Type – p. 43 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. 43 Multi-indexed polynomials

What about replacing pk(n) with pj,k(n)?

Polynomial Sequences of Binomial Type – p. 44 Multi-indexed polynomials

What about replacing pk(n) with pj,k(n)? To be investigated.

Polynomial Sequences of Binomial Type – p. 44 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).

xj yk χ (n) = (ex + ey − 1)n Kjk j! k! j,kX≥0

Polynomial Sequences of Binomial Type – p. 44 Multivariate polynomials

What about replacing pk(n) with pk(m, n)?

Polynomial Sequences of Binomial Type – p. 45 Multivariate polynomials

What about replacing pk(n) with pk(m, n)? Not yet considered.

May involve F (x)mG(x)n.

Polynomial Sequences of Binomial Type – p. 45 Continuous variant

Theorem (Schneider). Let S be a bounded measurable set in d-dimensional Euclidean d space. Let Pk(n ) 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 dk d n Pk(n ) is eventually a polynomial pk(n) for each k, and these polynomials form a sequence of binomial type.

Polynomial Sequences of Binomial Type – p. 46 A reference arXiv:1206.6174

Polynomial Sequences of Binomial Type – p. 47 The last slide

Polynomial Sequences of Binomial Type – p. 48 The last slide

Polynomial Sequences of Binomial Type – p. 49 The last slide

Polynomial Sequences of Binomial Type – p. 49