A072169 -id:A072169 - 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

A061256 Euler transform of sigma(n), cf. A000203.

Original entry on oeis.org

1, 1, 4, 8, 21, 39, 92, 170, 360, 667, 1316, 2393, 4541, 8100, 14824, 26071, 46422, 80314, 139978, 238641, 408201, 686799, 1156062, 1920992, 3189144, 5238848, 8589850, 13963467, 22641585, 36447544, 58507590, 93334008, 148449417, 234829969, 370345918

Offset: 0

Vladeta Jovovic, Apr 21 2001

This is also the number of ordered triples of permutations f, g, h in Symm(n) which all commute, divided by n!. This was conjectured by Franklin T. Adams-Watters, Jan 16 2006, and proved by J. R. Britnell in 2012.
According to a message on a blog page by "Allan" (see Secret Blogging Seminar link) it appears that a(n) = number of conjugacy classes of commutative ordered pairs in Symm(n).
John McKay (email to N. J. A. Sloane, Apr 23 2013) observes that A061256 and A006908 coincide for a surprising number of terms, and asks for an explanation. - N. J. A. Sloane, May 19 2013

			1 + x + 4*x^2 + 8*x^3 + 21*x^4 + 39*x^5 + 92*x^6 + 170*x^7 + 360*x^8 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
Lida Ahmadi, Ricardo Gómez Aíza, and Mark Daniel Ward, A unified treatment of families of partition functions, La Matematica (2024). Preprint available as arXiv:2303.02240 [math.CO], 2023.
J. R. Britnell, A formal identity involving commuting triples of permutations, arXiv:1203.5079 [math.CO], 2012.
J. R. Britnell, A formal identity involving commuting triples of permutations, Preprint 2012. - _N. J. A. Sloane_, Jun 13 2012
J. R. Britnell, A formal identity involving commuting triples of permutations, Journal of Combinatorial Theory, Series A, Volume 120, Issue 4, May 2013.
E. Marberg, How to compute the Frobenius-Schur indicator of a unipotent character of a finite Coxeter system, arXiv preprint arXiv:1202.1311 [math.RT], 2012. - _N. J. A. Sloane_, Jun 10 2012
Secret Blogging Seminar, A peculiar numerical coincidence.
N. J. A. Sloane, Transforms
Tad White, Counting Free Abelian Actions, arXiv:1304.2830 [math.CO], 2013.

Product_{k>=1} 1/(1 - x^k)^sigma_m(k): A006171 (m=0), this sequence (m=1), A275585 (m=2), A288391 (m=3), A301542 (m=4), A301543 (m=5), A301544 (m=6), A301545 (m=7), A301546 (m=8), A301547 (m=9).

Cf. A000203, A001001, A001970, A053529, A061255, A061257, A006908, A192065.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(
      d*sigma(d), d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Jun 08 2017

nn = 30; b = Table[DivisorSigma[1, n], {n, nn}]; CoefficientList[Series[Product[1/(1 - x^m)^b[[m]], {m, nn}], {x, 0, nn}], x] (* T. D. Noe, Jun 18 2012 *)
nmax = 40; CoefficientList[Series[Product[1/QPochhammer[x^k]^k, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 29 2015 *)

N=66; x='x+O('x^N); gf=1/prod(j=1,N, eta(x^j)^j); Vec(gf) /* Joerg Arndt, May 03 2008 */

{a(n)=if(n==0,1,polcoeff(exp(sum(m=1,n,sigma(m)*x^m/(1-x^m+x*O(x^n))^2/m)),n))} /* Paul D. Hanna, Mar 28 2009 */

a(n) = A072169(n) / n!.
G.f.: Product_{k=1..infinity} (1 - x^k)^(-sigma(k)). a(n)=1/n*Sum_{k=1..n} a(n-k)*b(k), n>1, a(0)=1, b(k)=Sum_{d|k} d*sigma(d), cf. A001001.
G.f.: exp( Sum_{n>=1} sigma(n)*x^n/(1-x^n)^2 /n ). [Paul D. Hanna, Mar 28 2009]
G.f.: exp( Sum_{n>=1} sigma_2(n)*x^n/(1-x^n)/n ). [Vladeta Jovovic, Mar 28 2009]
G.f.: prod(n>=1, E(x^n)^n ) where E(x) = prod(k>=1, 1-x^k). [Joerg Arndt, Apr 12 2013]
a(n) ~ exp((3*Pi)^(2/3) * Zeta(3)^(1/3) * n^(2/3)/2 - Pi^(4/3) * n^(1/3) / (4 * 3^(2/3) * Zeta(3)^(1/3)) - 1/24 - Pi^2/(288*Zeta(3))) * A^(1/2) * Zeta(3)^(11/72) / (2^(11/24) * 3^(47/72) * Pi^(11/72) * n^(47/72)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Mar 23 2018

Entry revised by N. J. A. Sloane, Jun 13 2012

A305127 Expansion of e.g.f. Product_{k>=1} 1/(1 - x^k)^(sigma(k)/k), where sigma(k) is the sum of the divisors of k.

Original entry on oeis.org

1, 1, 5, 23, 179, 1279, 13699, 135085, 1764377, 22527521, 344625461, 5283739471, 94562354875, 1685808248383, 33947023942259, 694786150879829, 15613612524749489, 357353282848083265, 8880505496901812197, 224851013929747732231, 6106205671049245677251, 169523515381173773551871

Offset: 0

Ilya Gutkovskiy, May 26 2018

nonn

a(n)/n! is the Euler transform of [1, 3/2, 4/3, 7/4, 6/5, ... = sums of reciprocals of divisors of 1, 2, 3, 4, 5, ...].

Vaclav Kotesovec, Table of n, a(n) for n = 0..438
Lida Ahmadi, Ricardo Gómez Aíza, and Mark Daniel Ward, A unified treatment of families of partition functions, La Matematica (2024). Preprint available as arXiv:2303.02240 [math.CO], 2023.
N. J. A. Sloane, Transforms

Cf. A000203, A007429, A017665, A017666, A028342, A053529, A061256, A072169, A318769, A318814.

with(numtheory): a := proc(n) option remember; `if`(n = 0, 1, add(add(sigma(d), d = divisors(j))*a(n-j), j = 1..n)/n) end proc; seq(n!*a(n), n = 0..20); # Vaclav Kotesovec, Sep 04 2018

nmax = 21; CoefficientList[Series[Product[1/(1 - x^k)^(DivisorSigma[1, k]/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[Exp[Sum[Sum[x^(j k)/(j k (1 - x^(j k))), {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d DivisorSigma[-1, d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[n! a[n], {n, 0, 21}]
nmax = 21; s = 1 - x; Do[s *= Sum[Binomial[DivisorSigma[1, k]/k, j]*(-1)^j*x^(j*k), {j, 0, nmax/k}]; s = Expand[s]; s = Take[s, Min[nmax + 1, Exponent[s, x] + 1, Length[s]]];, {k, 2, nmax}]; CoefficientList[Series[1/s, {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Sep 03 2018 *)

E.g.f.: Product_{k>=1} 1/(1 - x^k)^(A017665(k)/A017666(k)).
E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} x^(j*k)/(j*k*(1 - x^(j*k)))).
log(a(n)/n!) ~ sqrt(n) * Pi^2 / 3. - Vaclav Kotesovec, Sep 04 2018

A362827 Array read by antidiagonals: T(n,k) is the number of k-tuples of permutations of [n] that pairwise commute.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 6, 1, 1, 1, 8, 18, 24, 1, 1, 1, 16, 48, 120, 120, 1, 1, 1, 32, 126, 504, 840, 720, 1, 1, 1, 64, 336, 2016, 4680, 7920, 5040, 1, 1, 1, 128, 918, 7944, 24720, 66240, 75600, 40320, 1, 1, 1, 256, 2568, 31200, 130440, 516240, 856800, 887040, 362880, 1

Offset: 0

Andrew Howroyd, May 08 2023

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

			Array begins:
========================================================
n/k| 0    1     2      3       4        5          6 ...
---+----------------------------------------------------
0  | 1    1     1      1       1        1          1 ...
1  | 1    1     1      1       1        1          1 ...
2  | 1    2     4      8      16       32         64 ...
3  | 1    6    18     48     126      336        918 ...
4  | 1   24   120    504    2016     7944      31200 ...
5  | 1  120   840   4680   24720   130440     699840 ...
6  | 1  720  7920  66240  516240  3968640   30672720 ...
7  | 1 5040 75600 856800 9122400 97030080 1050336000 ...
  ...

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=0..3 are A000012, A000142, A053529, A072169.
Main diagonal is A362828.
Cf. A362824, A362826.

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
M(n,m=n)={my(v=vector(m+1), u=vector(n,n,n==1), f=vector(n,n,n!)); v[1]=vectorv(n+1,i,1); for(j=1, #v-1, my(t=EulerT(u)); v[j+1]=vectorv(n+1,i,i--;if(i,f[i]*t[i],1)); u=dirmul(u, vector(n, n, n^(j-1)))); Mat(v)}
{ my(A=M(7)); for(n=1, #A, print(A[n,])) }

T(n,k) = n!*A362826(n,k) for k > 0.

cp's OEIS Frontend

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

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A061256 Euler transform of sigma(n), cf. A000203.

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A305127 Expansion of e.g.f. Product_{k>=1} 1/(1 - x^k)^(sigma(k)/k), where sigma(k) is the sum of the divisors of k.

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A362827 Array read by antidiagonals: T(n,k) is the number of k-tuples of permutations of [n] that pairwise commute.

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