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A362826
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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.
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4
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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
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OFFSET
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0,9
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COMMENTS
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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.
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LINKS
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FORMULA
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Column k is the Euler transform of column k-1 of A160870.
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)).
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EXAMPLE
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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 ...
...
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PROG
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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), 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, ])) }
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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