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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.
5
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
0,9
COMMENTS
Two permutations x,y on [n] commute if x*y = y*x.
LINKS
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.
FORMULA
T(n,k) = n!*A362826(n,k) for k > 0.
EXAMPLE
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 ...
...
PROG
(PARI)
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, ])) }
CROSSREFS
Columns k=0..3 are A000012, A000142, A053529, A072169.
Main diagonal is A362828.
Sequence in context: A140274 A095231 A303697 * A342413 A202019 A295685
KEYWORD
nonn,tabl
AUTHOR
Andrew Howroyd, May 08 2023
STATUS
approved