cp's OEIS Frontend

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.

%I A246106 #39 Oct 05 2024 08:58:15
%S A246106 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,
%T A246106 175,8240,90492,5624,1,0,1,7,351,57675,7880456,64796982,251610,1,0,1,
%U A246106 8,637,289716,270656150,79846389608,302752867740,33642660,1,0
%N 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.
%H A246106 Alois P. Heinz, <a href="/A246106/b246106.txt">Antidiagonals n = 0..27, flattened</a>
%H A246106 <a href="/index/Mat#inequiv">Index to number of inequivalent matrices modulo permutation of rows and columns</a>
%F A246106 A(n,k) = Sum_{i=0..k} C(k,i) * A256069(n,i).
%F A246106 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]
%e A246106 Square array A(n,k) begins:
%e A246106   1, 1,    1,        1,           1,              1, ...
%e A246106   0, 1,    2,        3,           4,              5, ...
%e A246106   0, 1,    7,       27,          76,            175, ...
%e A246106   0, 1,   36,      738,        8240,          57675, ...
%e A246106   0, 1,  317,    90492,     7880456,      270656150, ...
%e A246106   0, 1, 5624, 64796982, 79846389608, 20834113243925, ...
%p A246106 b:= proc(n, i) option remember; `if`(n=0, [[]],
%p A246106       `if`(i<1, [], [b(n, i-1)[], seq(map(p->[p[], [i, j]],
%p A246106        b(n-i*j, i-1))[], j=1..n/i)]))
%p A246106     end:
%p A246106 A:= proc(n, k) option remember; add(add(k^add(add(i[2]*j[2]*
%p A246106       igcd(i[1], j[1]), j=t), i=s) /mul(i[1]^i[2]*i[2]!, i=s)
%p A246106       /mul(i[1]^i[2]*i[2]!, i=t), t=b(n$2)), s=b(n$2))
%p A246106     end:
%p A246106 seq(seq(A(n, d-n), n=0..d), d=0..10);
%o A246106 (PARI) A246106(n,k)=A353585(k,n,n) \\ _M. F. Hasler_, May 01 2022
%Y A246106 Columns k = 0-10 give: A000007, A000012, A002724, A052269, A052271, A052272, A246112, A246113, A246114, A246115, A246116.
%Y A246106 Rows n = 0-10 give: A000012, A001477, A039623, A058001, A058002, A058003, A058004, A246108, A246109, A246110, A246111.
%Y A246106 Main diagonal gives A246107.
%Y A246106 A028657, A242106, A353585 are related tables.
%Y A246106 Cf. A242095, A256069.
%K A246106 nonn,tabl
%O A246106 0,8
%A A246106 _Alois P. Heinz_, Aug 13 2014