A304942 - OEIS

%I #24 Aug 29 2018 20:42:35

%S 1,1,1,1,2,1,1,3,3,1,1,5,11,5,1,1,7,35,35,7,1,1,11,132,410,132,11,1,1,

%T 15,471,6178,6178,471,15,1,1,22,1806,122038,594203,122038,1806,22,1,1,

%U 30,7042,2921607,85820809,85820809,2921607,7042,30,1

%N Triangle read by rows: T(n,k) is the number of nonisomorphic binary n X n matrices with k 1's per column under row and column permutations.

%H Andrew Howroyd, <a href="/A304942/b304942.txt">Table of n, a(n) for n = 0..527</a>

%H StackExchange, <a href="http://math.stackexchange.com/questions/616834/">How many arrays with crossed cells, order of rows/columns irrelevant</a>, Dec 13 2013

%e Triangle begins (n >=0, k >= 0):

%e   1;

%e   1,  1;

%e   1,  2,    1;

%e   1,  3,    3,      1;

%e   1,  5,   11,      5,      1;

%e   1,  7,   35,     35,      7,      1;

%e   1, 11,  132,    410,    132,     11,    1;

%e   1, 15,  471,   6178,   6178,    471,   15,  1;

%e   1, 22, 1806, 122038, 594203, 122038, 1806, 22, 1;

%e   ...

%o (PARI)

%o permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}

%o K(q,t,k)={polcoeff(prod(j=1, #q, my(g=gcd(t, q[j])); (1 + x^(q[j]/g) + O(x*x^k))^g), k)}

%o Blocks(n,m,k)={my(s=0); forpart(q=m, s+=permcount(q)*polcoeff(exp(sum(t=1, n, K(q,t,k)/t*x^t) + O(x*x^n)), n)); s/m!}

%o for(n=0, 10, for(k=0, n, print1(Blocks(n,n,k), ", ")); print)

%Y Columns k=1..5 are A000041, A247417, A247596, A247597, A247598.

%Y Cf. A305027.

%K nonn,tabl

%O 0,5

%A _Andrew Howroyd_, May 23 2018