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

0,3

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

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

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 *)

Prog

(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
(Magma) a:= func< n | NumberOfPartitions(n)*Factorial(n) >; [ a(n) : n in [0..25]]; // Vincenzo Librandi, Jan 17 2019
(Python)
from math import factorial
from sympy import npartitions
def A053529(n): return factorial(n)*npartitions(n) # Chai Wah Wu, Jul 10 2023

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.

Sequence in context: A321704 A296982 A222375 * A005442 A306881 A367489

Adjacent sequences: A053526 A053527 A053528 * A053530 A053531 A053532

Keyword

nonn,easy,nice

Author

N. J. A. Sloane, Jan 16 2000

Status

approved