A092552 Let X_{m,n}(q) be the chromatic polynomial of the complete bipartite graph K_{m,n}. Then a(n) is the negative of the coefficient of the linear term of X_{n,n}(q).

0, 1, 3, 31, 675, 25231, 1441923, 116914351, 12764590275, 1805409270031, 321113303226243, 70146437009397871, 18462286083671614275, 5762225835975165678031, 2104263061425865873128963, 888881838896989670838028591, 430058409024841744606172532675

Offset

0,3

Comments

From Arvind Ayyer, Oct 25 2020: (Start)

Equivalently, a(n) is the number of acyclic orientations with a unique sink in K_{n,n}.

a(n) is also the number of toppleable permutations in S_{2n-1}. A toppleable permutation pi in S_{2n-1} satisfies pi_i <= n-1+i for 1 <= i <= n-1 and pi_i >= i-n+1 for n <= i <= 2n-1. The a(3)=3 toppleable permutations in S_3 are 123, 213 and 132. (End)

Links

Alois P. Heinz, Table of n, a(n) for n = 0..238

A. Ayyer, D. Hathcock and P. Tetali, Toppleable Permutations, Excedances and Acyclic Orientations, arXiv:2010.11236 [math.CO], 2020. For the precise definition of a toppleable permutation.

Formula

From Alois P. Heinz, Apr 30 2012: (Start)

a(n) = (-1) * [q] Sum_{j=1..n} (q-j)^n*S2(n,j)*Product_{i=0..j-1} (q-i).

a(n) = (-1) * A212084(n,2n-1). (End)

E.g.f.: Sum_{n>=1} (1 - exp(-n*x))^n / n. - Paul D. Hanna, Dec 06 2012

a(n) = Sum_{k=1..n} k!*(k-1)! * Stirling2(n, k)^2. - Paul D. Hanna, Dec 30 2012, corrected by Vaclav Kotesovec, Jun 21 2013

O.g.f.: Sum_{n>=1} n^(n-1) * n! * x^n / Product_{k=1..n} (1 + n*k*x). - Paul D. Hanna, Jan 05 2013

a(n) = A136126(2*n-1,n), where triangle A136126(n,k) is the number of permutations of {1,2,...,k+n} having excedance set {1,2,...,k}. - Paul D. Hanna, Feb 01 2013

a(n) ~ sqrt(Pi) * n^(2*n-1/2) / (sqrt(1-log(2)) * exp(2*n) * (log(2))^(2*n)). - Vaclav Kotesovec, Nov 07 2014

a(n) = A306209(2n-1,n-1) for n > 0. - Alois P. Heinz, Feb 01 2019

Example

a(2) = 3 since the chromatic polynomial of K_{2,2}(q) is q^4-4*q^3+6*q^2-3*q.
E.g.f.: A(x) = x + 3*x^2/2! + 31*x^3/3! + 675*x^4/4! + 25231*x^5/5! +...
where A(x) = (1-exp(-x)) + (1-exp(-2*x))^2/2 + (1-exp(-3*x))^3/3 +... - Paul D. Hanna, Dec 06 2012
O.g.f.: F(x) = x + 3*x^2 + 31*x^3 + 675*x^4 + 25231*x^5 +...
where F(x) = x/(1+x) + 2^1*2!*x^2/((1+2*1*x)*(1+2*2*x)) + 3^2*3!*x^3/((1+3*1*x)*(1+3*2*x)*(1+3*3*x)) + 4^3*4!*x^4/((1+4*1*x)*(1+4*2*x)*(1+4*3*x)*(1+4*4*x)) +... - Paul D. Hanna, Jan 05 2013

Maple

a:= n-> -coeff(add(Stirling2(n, k) *mul(q-i, i=0..k-1)
             *(q-k)^n, k=1..n), q, 1):
seq(a(n), n=0..20);  # Alois P. Heinz, Apr 30 2012

Mathematica

Table[Sum[k!*(k-1)!*StirlingS2[n, k]^2, {k, 1, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jun 21 2013 *)

Prog

(PARI) {a(n)=n!*polcoeff(sum(k=1, n, (1-exp(-k*x+x*O(x^n)))^k/k), n)} \\ Paul D. Hanna, Dec 06 2012
for(n=0, 20, print1(a(n), ", "))
(PARI) {Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n)=if(n<=0, 0, sum(k=1, n, k!*(k-1)! * Stirling2(n, k)^2))} \\ Paul D. Hanna, Dec 30 2012
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n)=if(n<1, 0, polcoeff(sum(m=1, n, m^(m-1)*m!*x^m/prod(k=1, m, 1+m*k*x+x*O(x^n))), n))} \\ Paul D. Hanna, Jan 05 2013

Crossrefs

Cf. A008277, A212084, A220179, A136126, A306209.

Sequence in context: A105293 A368489 A104841 * A322487 A300735 A196457

Adjacent sequences: A092549 A092550 A092551 * A092553 A092554 A092555

Keyword

nonn

Author

Michael Lugo (mtlugo(AT)mit.edu), Apr 09 2004

Status

approved