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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. 4
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,9 COMMENTS 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. 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 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)). EXAMPLE 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 ... ... 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), 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, ])) } CROSSREFS Columns k=1..4 are A000012, A000041, A061256, A226313. Cf. A160870, A362827, A362903. Sequence in context: A321312 A327502 A354090 * A220632 A125653 A104445 Adjacent sequences: A362823 A362824 A362825 * A362827 A362828 A362829 KEYWORD nonn,tabl AUTHOR Andrew Howroyd, May 09 2023 STATUS approved

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Last modified August 12 15:11 EDT 2024. Contains 375113 sequences. (Running on oeis4.)