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a(n) = n! * number of partitions of n.
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%I #58 Jul 10 2023 12:14:27

%S 1,1,4,18,120,840,7920,75600,887040,10886400,152409600,2235340800,

%T 36883123200,628929100800,11769069312000,230150688768000,

%U 4833164464128000,105639166144512000,2464913876705280000,59606099200327680000,1525429559126753280000,40464026199993876480000

%N a(n) = n! * number of partitions of n.

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

%C Equivalently sum of the order of all normalizers of all cyclic subgroups of Sym(n). - _Olivier GĂ©rard_, Apr 04 2012

%C From _Gus Wiseman_, Jan 16 2019: (Start)

%C 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:

%C 123 213 132 312 231 321

%C .

%C 12 21 13 31 23 32

%C 3 3 2 2 1 1

%C .

%C 1 2 1 3 2 3

%C 2 1 3 1 3 2

%C 3 3 2 2 1 1

%C (End)

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

%H T. D. Noe, <a href="/A053529/b053529.txt">Table of n, a(n) for n = 0..200</a>

%H M. Holloway, M. Shattuck, <a href="http://puma.dimai.unifi.it/24_1/2.holloway_shattuck.pdf">Commuting pairs of functions on a finite set</a>, PU.M.A, Volume 24 (2013), Issue No. 1.

%H M. Holloway, M. Shattuck, <a href="http://www.researchgate.net/profile/Mark_Shattuck/publication/272492907">Commuting pairs of functions on a finite set</a>, Research Gate, 2015.

%H R. P. Stanley, <a href="http://www.jstor.org/stable/2589191">Pairs with equal squares, Problem 10654</a>, Amer. Math. Monthly, 107 (April 2000), solution p. 368.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Young_tableau">Young tableau</a>

%F 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

%F 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

%F a(n) ~ sqrt(Pi/6)*exp(sqrt(2/3)*Pi*sqrt(n))*n^n/(2*exp(n)*sqrt(n)). - _Ilya Gutkovskiy_, Jul 28 2016

%p seq(count(Permutation(n))*count(Partition(n)),n=1..20); # _Zerinvary Lajos_, Oct 16 2006

%p with(combinat): A053529 := proc(n): n! * numbpart(n) end: seq(A053529(n), n=0..20); # _Johannes W. Meijer_, Jul 28 2016

%t Table[PartitionsP[n] n!, {n, 0, 20}] (* _T. D. Noe_, Jun 19 2012 *)

%o (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

%o (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

%o (PARI) a(n) = n!*numbpart(n); \\ _Michel Marcus_, Jul 28 2016

%o (Magma) a:= func< n | NumberOfPartitions(n)*Factorial(n) >; [ a(n) : n in [0..25]]; // _Vincenzo Librandi_, Jan 17 2019

%o (Python)

%o from math import factorial

%o from sympy import npartitions

%o def A053529(n): return factorial(n)*npartitions(n) # _Chai Wah Wu_, Jul 10 2023

%Y Column k=2 of A362827.

%Y Cf. A000041, A072169, A061256.

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

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

%K nonn,easy,nice

%O 0,3

%A _N. J. A. Sloane_, Jan 16 2000