A246106 Number A(n,k) of inequivalent n X n matrices with entries from [k], where equivalence means permutations of rows or columns; square array A(n,k), n>=0, k>=0, read by antidiagonals.
1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 7, 1, 0, 1, 4, 27, 36, 1, 0, 1, 5, 76, 738, 317, 1, 0, 1, 6, 175, 8240, 90492, 5624, 1, 0, 1, 7, 351, 57675, 7880456, 64796982, 251610, 1, 0, 1, 8, 637, 289716, 270656150, 79846389608, 302752867740, 33642660, 1, 0
Offset: 0
Examples
Square array A(n,k) begins: 1, 1, 1, 1, 1, 1, ... 0, 1, 2, 3, 4, 5, ... 0, 1, 7, 27, 76, 175, ... 0, 1, 36, 738, 8240, 57675, ... 0, 1, 317, 90492, 7880456, 270656150, ... 0, 1, 5624, 64796982, 79846389608, 20834113243925, ...
Links
Crossrefs
Programs
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Maple
b:= proc(n, i) option remember; `if`(n=0, [[]], `if`(i<1, [], [b(n, i-1)[], seq(map(p->[p[], [i, j]], b(n-i*j, i-1))[], j=1..n/i)])) end: A:= proc(n, k) option remember; add(add(k^add(add(i[2]*j[2]* igcd(i[1], j[1]), j=t), i=s) /mul(i[1]^i[2]*i[2]!, i=s) /mul(i[1]^i[2]*i[2]!, i=t), t=b(n$2)), s=b(n$2)) end: seq(seq(A(n, d-n), n=0..d), d=0..10); -
PARI
A246106(n,k)=A353585(k,n,n) \\ M. F. Hasler, May 01 2022
Formula
A(n,k) = Sum_{i=0..k} C(k,i) * A256069(n,i).
A(n,k) = Sum_{p,q in P(n)} k^Sum_{i in p, j in q} gcd(i, j) / (N(p)*N(q)) where N(p) = Product_{distinct parts x in p} x^m(x)*m(x)!, m(x) = multiplicity of x in p. - M. F. Hasler, Apr 30 2022 [corrected by Anders Kaseorg, Oct 04 2024]
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