A053529 a(n) = n! * number of partitions of n.

1, 1, 4, 18, 120, 840, 7920, 75600, 887040, 10886400, 152409600, 2235340800, 36883123200, 628929100800, 11769069312000, 230150688768000, 4833164464128000, 105639166144512000, 2464913876705280000, 59606099200327680000, 1525429559126753280000, 40464026199993876480000

Offset: 0

Author

N. J. A. Sloane, Jan 16 2000

Keywords

• nonn
• easy
• nice

Comments

Commuting permutations: number of ordered pairs (g, h) in Sym(n) such that gh = hg.

Equivalently sum of the order of all normalizers of all cyclic subgroups of Sym(n). - Olivier Gérard, Apr 04 2012

From Gus Wiseman, Jan 16 2019: (Start)

Also the number of Young tableaux with distinct entries from 1 to n, where a Young tableau is an array obtained by replacing the dots in the Ferrers diagram of an integer partition of n with positive integers. For example, the a(3) = 18 tableaux are:

123 213 132 312 231 321

.

12 21 13 31 23 32

3 3 2 2 1 1

.

1 2 1 3 2 3

2 1 3 1 3 2

3 3 2 2 1 1

(End)

References

• R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.12, solution.

Links

• T. D. Noe, Table of n, a(n) for n = 0..200
• M. Holloway, M. Shattuck, Commuting pairs of functions on a finite set, PU.M.A, Volume 24 (2013), Issue No. 1.
• M. Holloway, M. Shattuck, Commuting pairs of functions on a finite set, Research Gate, 2015.
• R. P. Stanley, Pairs with equal squares, Problem 10654, Amer. Math. Monthly, 107 (April 2000), solution p. 368.
• Wikipedia, Young tableau

Crossrefs

Column k=2 of A362827.

Cf. A000041, A072169, A061256.

Sequences counting pairs of functions from an n-set to itself: A053529, A181162, A239749-A239785, A239836-A239841.

Cf. A000085, A117433, A153452, A296188, A323295, A323434, A323436.

Programs

Magma

a:= func< n | NumberOfPartitions(n)*Factorial(n) >; [ a(n) : n in [0..25]]; // Vincenzo Librandi, Jan 17 2019

Maple

seq(count(Permutation(n))*count(Partition(n)),n=1..20); # Zerinvary Lajos, Oct 16 2006
with(combinat): A053529 := proc(n): n! * numbpart(n) end: seq(A053529(n), n=0..20); # Johannes W. Meijer, Jul 28 2016

Mathematica

Table[PartitionsP[n] n!, {n, 0, 20}] (* T. D. Noe, Jun 19 2012 *)

PARI

N=66; x='x+O('x^N); Vec(serlaplace(exp(sum(k=1, N, x^k/(1-x^k)/k)))) \\ Joerg Arndt, Apr 16 2010

PARI

N=66; x='x+O('x^N); Vec(serlaplace(sum(n=0, N, x^n/prod(k=1,n,1-x^k)))) \\ Joerg Arndt, Jan 29 2011

PARI

a(n) = n!*numbpart(n); \\ Michel Marcus, Jul 28 2016

Python

from math import factorial
from sympy import npartitions
def A053529(n): return factorial(n)*npartitions(n) # Chai Wah Wu, Jul 10 2023

Formula

E.g.f: Sum_{n>=0} x^n/(Product_{k=1..n} 1-x^k) = exp(Sum_{n>=1} (x^n/n)/(1-x^n)). - Joerg Arndt, Jan 29 2011

a(n) = Sum{k=1..n} (((n-1)!/(n-k)!)*sigma(k)*a(n-k)), n > 0, and a(0)=1. See A274760. - Johannes W. Meijer, Jul 28 2016

a(n) ~ sqrt(Pi/6)*exp(sqrt(2/3)*Pi*sqrt(n))*n^n/(2*exp(n)*sqrt(n)). - Ilya Gutkovskiy, Jul 28 2016