A362826 -id:A362826 - cp's OEIS frontend

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

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.

A226313 Number of commuting 4-tuples of elements from S_n, divided by n!.

Original entry on oeis.org

1, 8, 21, 84, 206, 717, 1810, 5462, 13859, 38497, 96113, 253206, 620480, 1566292, 3770933, 9212041, 21768608, 51795427, 120279052, 279849177, 639379257, 1459282932, 3283758256, 7369471795, 16351101855, 36147590987, 79162129897, 172646751524, 373527250619, 804631686843, 1721283389932, 3666041417241

Offset: 1

N. J. A. Sloane, Jun 08 2013

nonn

Euler transform of A001001.

Alois P. Heinz, Table of n, a(n) for n = 1..9000
Tad White, Counting Free Abelian Actions, arXiv preprint arXiv:1304.2830, 2013

Column k=4 of A362826.

Cf. A001001, A061256, A301777.

with(numtheory):
b:= proc(n) option remember; add(d*sigma(d), d=divisors(n)) end:
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      b(d), d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=1..40);  # Alois P. Heinz, Mar 06 2015

b[n_] := b[n] = DivisorSum[n, #*DivisorSigma[1, #]&];
a[n_] := a[n] = If[n == 0, 1, Sum[DivisorSum[j, #*b[#]&]*a[n-j], {j, 1, n}] /n];
Array[a, 40] (* Jean-François Alcover, Mar 27 2017, after Alois P. Heinz *)
nmax = 40; Rest[CoefficientList[Series[Exp[Sum[Sum[Sum[d*DivisorSigma[1, d], {d, Divisors[k]}] * x^(j*k) / j, {k, 1, Floor[nmax/j] + 1}], {j, 1, nmax}]], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Mar 31 2018 *)

EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
seq(n)={my(v=vector(n,i,1)); for(k=1, 2, v=dirmul(v, vector(n,i,i^k))); EulerT(v)} \\ Andrew Howroyd, May 09 2023

a(n) ~ exp(2^(7/4) * Pi^(3/2) * Zeta(3)^(1/4) * n^(3/4) / (3^(3/2) * 5^(1/4)) - sqrt(5*Zeta(3)*n) / (2^(3/2)*Pi) + (sqrt(Pi) * 5^(1/4) / (2^(15/4) * 3^(3/2) * Zeta(3)^(1/4)) - sqrt(3) * 5^(5/4) * Zeta(3)^(3/4) / (2^(15/4) * Pi^(7/2))) * n^(1/4) - 25*Zeta(3) / (16*Pi^6) + (5 - 2*Zeta(3)) / (192*Pi^2)) * Pi^(1/4) * Zeta(3)^(1/8) / (2^(13/8) * 3^(1/4) * 5^(1/8) * n^(5/8)). - Vaclav Kotesovec, Mar 26 2018

A362903 Array read by antidiagonals: T(n,k) is the number of nonisomorphic k-tuples of involutions on a (2n)-set that pairwise commute.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 8, 11, 4, 1, 1, 16, 43, 24, 5, 1, 1, 32, 171, 176, 46, 6, 1, 1, 64, 683, 1376, 611, 80, 7, 1, 1, 128, 2731, 10944, 9281, 1864, 130, 8, 1, 1, 256, 10923, 87424, 146445, 54384, 5161, 200, 9, 1, 1, 512, 43691, 699136, 2334181, 1696352, 285939, 13184, 295, 10, 1

Offset: 0

Andrew Howroyd, May 11 2023

Two involutions x,y commute if x*y = y*x. Isomorphism is up to permutation of the elements of the (2n)-set. T(n,k) also gives the values for a (2n+1)-set.

			Array begins:
======================================================
n/k| 0 1   2     3       4          5            6 ...
---+--------------------------------------------------
0  | 1 1   1     1       1          1            1 ...
1  | 1 2   4     8      16         32           64 ...
2  | 1 3  11    43     171        683         2731 ...
3  | 1 4  24   176    1376      10944        87424 ...
4  | 1 5  46   611    9281     146445      2334181 ...
5  | 1 6  80  1864   54384    1696352     53885632 ...
6  | 1 7 130  5161  285939   17562679   1110290303 ...
7  | 1 8 200 13184 1372224  165343616  20774749952 ...
8  | 1 9 295 31532 6101080 1436647664 358238974304 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (first 51 antidiagonals)

Columns k=0..3 are A000012, A000027(n-1), A001752, A362904.
Rows n=1..3 are A000079, A007583, A103334(n+1).
Cf. A022166, A362648, A362824, A362826.

\\ B(n, k) is A022166.
B(n, k)={polcoef(x^k/prod(j=0, k, 1-2^j*x + O(x*x^n)), n)}
C(k,n) = Vec(1/prod(j=0, min(k-1, logint(n, 2)), (1 - x^(2^j) + O(x*x^n))^B(k,j+1), 1 - x + O(x*x^n)))
M(n,m=n) = Mat(vector(m+1, k, C(k-1, n)~))
{ my(A=M(7)); for(i=1, #A, print(A[i,])) }

G.f. of column k: 1/((1 - x)*Product_{j=0..k-1} (1 - x^(2^j))^A022166(k,j+1)).

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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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A226313 Number of commuting 4-tuples of elements from S_n, divided by n!.

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A362903 Array read by antidiagonals: T(n,k) is the number of nonisomorphic k-tuples of involutions on a (2n)-set that pairwise commute.

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