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%I #13 Jan 25 2020 17:55:35
%S 1,1,1,1,1,1,1,2,0,1,1,4,3,0,1,1,8,23,0,0,1,1,16,290,184,0,0,1,1,32,
%T 4298,17488,840,0,0,1,1,64,79143,2780752,771305,0,0,0,1,1,128,1702923,
%U 689187720,1496866413,21770070,0,0,0,1,1,256,42299820,236477490418,5261551562405,585897733896,328149360,0,0,0,1
%N Array read by antidiagonals: A(n,k) is the number of binary matrices with k columns and any number of distinct nonzero rows with n ones in every column and columns in nonincreasing lexicographic order.
%C The condition that the columns be in nonincreasing order is equivalent to considering nonequivalent matrices up to permutation of columns.
%H Andrew Howroyd, <a href="/A331571/b331571.txt">Table of n, a(n) for n = 0..209</a>
%F A(n, k) = Sum_{j=0..k} abs(Stirling1(k, j))*A331567(n, j)/k!.
%F A(n, k) = Sum_{j=0..k} binomial(k-1, k-j)*A331569(n, j).
%F A(n, k) = 0 for k > 0, n > 2^(k-1).
%F A331653(n) = Sum_{d|n} A(n/d, d).
%e Array begins:
%e ===============================================================
%e n\k | 0 1 2 3 4 5 6
%e ----+----------------------------------------------------------
%e 0 | 1 1 1 1 1 1 1 ...
%e 1 | 1 1 2 4 8 16 32 ...
%e 2 | 1 0 3 23 290 4298 79143 ...
%e 3 | 1 0 0 184 17488 2780752 689187720 ...
%e 4 | 1 0 0 840 771305 1496866413 5261551562405 ...
%e 5 | 1 0 0 0 21770070 585897733896 30607728081550686 ...
%e 6 | 1 0 0 0 328149360 161088785679360 ...
%e ...
%e The A(2,2) = 3 matrices are:
%e [1 1] [1 0] [1 0]
%e [1 0] [1 1] [0 1]
%e [0 1] [0 1] [1 1]
%o (PARI)
%o WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)}
%o D(p, n, k)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); binomial(WeighT(v)[n] + k - 1, k)/prod(i=1, #v, i^v[i]*v[i]!)}
%o T(n, k)={ my(m=n*k+1, q=Vec(exp(intformal(O(x^m) - x^n/(1-x)))), f=Vec(serlaplace(1/(1+x) + O(x*x^m))/(x-1))); if(n==0, 1, sum(j=1, m, my(s=0); forpart(p=j, s+=(-1)^#p*D(p, n, k), [1, n]); s*sum(i=j, m, q[i-j+1]*f[i]))); }
%Y Rows n=0..4 are A000012, A011782, A060090, A060491, A331652.
%Y Cf. A330942, A331567, A331569, A331570, A331572, A331653.
%K nonn,tabl
%O 0,8
%A _Andrew Howroyd_, Jan 20 2020
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