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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.

%I A362827 #11 May 09 2023 15:36:28
%S A362827 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,
%T A362827 32,126,504,840,720,1,1,1,64,336,2016,4680,7920,5040,1,1,1,128,918,
%U A362827 7944,24720,66240,75600,40320,1,1,1,256,2568,31200,130440,516240,856800,887040,362880,1
%N A362827 Array read by antidiagonals: T(n,k) is the number of k-tuples of permutations of [n] that pairwise commute.
%C A362827 Two permutations x,y on [n] commute if x*y = y*x.
%H A362827 Andrew Howroyd, <a href="/A362827/b362827.txt">Table of n, a(n) for n = 0..1325</a> (first 51 antidiagonals).
%H A362827 Tad White, <a href="http://arxiv.org/abs/1304.2830">Counting Free Abelian Actions</a>, arXiv preprint arXiv:1304.2830 [math.CO], 2013.
%F A362827 T(n,k) = n!*A362826(n,k) for k > 0.
%e A362827 Array begins:
%e A362827 ========================================================
%e A362827 n/k| 0    1     2      3       4        5          6 ...
%e A362827 ---+----------------------------------------------------
%e A362827 0  | 1    1     1      1       1        1          1 ...
%e A362827 1  | 1    1     1      1       1        1          1 ...
%e A362827 2  | 1    2     4      8      16       32         64 ...
%e A362827 3  | 1    6    18     48     126      336        918 ...
%e A362827 4  | 1   24   120    504    2016     7944      31200 ...
%e A362827 5  | 1  120   840   4680   24720   130440     699840 ...
%e A362827 6  | 1  720  7920  66240  516240  3968640   30672720 ...
%e A362827 7  | 1 5040 75600 856800 9122400 97030080 1050336000 ...
%e A362827   ...
%o A362827 (PARI)
%o A362827 EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
%o A362827 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)}
%o A362827 { my(A=M(7)); for(n=1, #A, print(A[n,])) }
%Y A362827 Columns k=0..3 are A000012, A000142, A053529, A072169.
%Y A362827 Main diagonal is A362828.
%Y A362827 Cf. A362824, A362826.
%K A362827 nonn,tabl
%O A362827 0,9
%A A362827 _Andrew Howroyd_, May 08 2023