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
Examples
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 ... ...
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.
Crossrefs
Programs
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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,])) }
Formula
T(n,k) = n!*A362826(n,k) for k > 0.
Comments