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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 (list; table; graph; refs; listen; history; text; internal format)
 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. Cf. A362824, A362826. Sequence in context: A140274 A095231 A303697 * A342413 A202019 A295685 Adjacent sequences: A362824 A362825 A362826 * A362828 A362829 A362830 KEYWORD nonn,tabl AUTHOR Andrew Howroyd, May 08 2023 STATUS approved

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Last modified August 3 20:33 EDT 2024. Contains 374905 sequences. (Running on oeis4.)