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A053529 a(n) = n! * number of partitions of n. 34
1, 1, 4, 18, 120, 840, 7920, 75600, 887040, 10886400, 152409600, 2235340800, 36883123200, 628929100800, 11769069312000, 230150688768000, 4833164464128000, 105639166144512000, 2464913876705280000, 59606099200327680000, 1525429559126753280000 (list; graph; refs; listen; history; text; internal format)
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

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.

FORMULA

E.g.f: sum(n>=0, x^n/prod(k=1,n,1-x^k)) = exp(sum(n>=1, 1/n*x^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

CROSSREFS

Cf. A000041, A072169, A061256.

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

Sequence in context: A321704 A296982 A222375 * A005442 A084661 A112294

Adjacent sequences:  A053526 A053527 A053528 * A053530 A053531 A053532

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane, Jan 16 2000

STATUS

approved

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Last modified November 19 10:48 EST 2018. Contains 317349 sequences. (Running on oeis4.)