A362827 -id:A362827 - cp's OEIS frontend

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

N. J. A. Sloane, Jan 16 2000

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)

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

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

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.

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

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

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

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

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

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

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

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

A072169 Commuting permutations: number of ordered triples of permutations f, g, h in Symm(n) which all commute.

Original entry on oeis.org

1, 1, 8, 48, 504, 4680, 66240, 856800, 14515200, 242040960, 4775500800, 95520902400, 2175146265600, 50438868480000, 1292330988748800, 34092378448128000, 971277752180736000, 28566680100102144000, 896191466580393984000, 29029508406664077312000

Offset: 0

N. J. A. Sloane, Sep 06 2003. More terms from A061256 from N. J. A. Sloane, Jun 13 2012

nonn

a(1)-a(7) computed by John McKay, Sep 06 2003.

T. D. Noe, Table of n, a(n) for n = 0..200

Column k=3 of A362827.

Cf. A061256, A053529.

for n in {1 .. 5} do G := SymmetricGroup(n); t1 := 0; for g in G do for h in G do for i in G do if g*h eq h*g and g*i eq i*g and h*i eq i*h then t1 := t1+1; end if; end for; end for; end for; n, t1; end for;

nn = 20; b = Table[DivisorSigma[1, n], {n, nn}]; Range[0, nn]! CoefficientList[Series[Product[1/(1 - x^m)^b[[m]], {m, nn}], {x, 0, nn}],  x] (* T. D. Noe, Jun 19 2012 *)

Equals A061256(n)*n!.

A362826 Array read by antidiagonals: T(n,k) is the number of k-tuples of permutations of [n] which commute, divided by n!, n >= 0, k >= 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 3, 1, 1, 1, 8, 8, 5, 1, 1, 1, 16, 21, 21, 7, 1, 1, 1, 32, 56, 84, 39, 11, 1, 1, 1, 64, 153, 331, 206, 92, 15, 1, 1, 1, 128, 428, 1300, 1087, 717, 170, 22, 1, 1, 1, 256, 1221, 5111, 5832, 5512, 1810, 360, 30, 1

Offset: 0

Andrew Howroyd, May 09 2023

T(n,k) is also the number of nonisomorphic (k-1)-tuples of permutations of an n-set that pairwise commute. Isomorphism is up to permutation of the elements of the n-set.

			Array begins:
=======================================================
n/k| 1  2   3    4     5       6        7         8 ...
---+---------------------------------------------------
0  | 1  1   1    1     1       1        1         1 ...
1  | 1  1   1    1     1       1        1         1 ...
2  | 1  2   4    8    16      32       64       128 ...
3  | 1  3   8   21    56     153      428      1221 ...
4  | 1  5  21   84   331    1300     5111     20144 ...
5  | 1  7  39  206  1087    5832    31949    178486 ...
6  | 1 11  92  717  5512   42601   333012   2635637 ...
7  | 1 15 170 1810 19252  208400  2303310  25936170 ...
8  | 1 22 360 5462 81937 1241302 19107225 299002252 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (first 51 antidiagonals).
Tad White, Counting Free Abelian Actions, arXiv preprint arXiv:1304.2830 [math.CO], 2013.

Columns k=1..4 are A000012, A000041, A061256, A226313.

Cf. A160870, A362827, A362903.

EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
M(n, m=n)={my(v=vector(m), u=vector(n, n, n==1)); for(j=1, #v, v[j]=concat([1], EulerT(u))~; u=dirmul(u, vector(n, n, n^(j-1)))); Mat(v)}
{ my(A=M(8)); for(n=1, #A~, print(A[n, ])) }

Column k is the Euler transform of column k-1 of A160870.
T(n,k) = A362827(n,k) / n!.
G.f. of column k: exp(Sum_{i>=1} x^i*A160870(i,k)/i).
G.f. of column k > 1: 1/(Product_{i>=1} (1 - x^i)^A160870(i,k-1)).

A362828 Number of n-tuples of permutations of [n] that pairwise commute.

Original entry on oeis.org

1, 1, 4, 48, 2016, 130440, 30672720, 11608682400, 12055770800640, 24154259257215360, 117792549941415955200, 1161512734993746635808000, 25629823970496421449477580800, 1215203193235691517749414518195200, 123585796012441765074167804498857267200

Offset: 0

Andrew Howroyd, May 08 2023

nonn

Two permutations x,y on [n] commute if x*y = y*x.

Andrew Howroyd, Table of n, a(n) for n = 0..50
Tad White, Counting Free Abelian Actions, arXiv preprint arXiv:1304.2830 [math.CO], 2013.

Main diagonal of A362827.

Cf. A362823.

cp's OEIS Frontend

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

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A072169 Commuting permutations: number of ordered triples of permutations f, g, h in Symm(n) which all commute.

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A362826 Array read by antidiagonals: T(n,k) is the number of k-tuples of permutations of [n] which commute, divided by n!, n >= 0, k >= 1.

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A362828 Number of n-tuples of permutations of [n] that pairwise commute.

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