A331567 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.
1, 1, 1, 1, 1, 1, 1, 3, 0, 1, 1, 13, 6, 0, 1, 1, 75, 120, 0, 0, 1, 1, 541, 6174, 1104, 0, 0, 1, 1, 4683, 449520, 413088, 5040, 0, 0, 1, 1, 47293, 49686726, 329520720, 18481080, 0, 0, 0, 1, 1, 545835, 7455901320, 491236986720, 179438982360, 522481680, 0, 0, 0, 1
Offset: 0
Examples
Array begins: =============================================================== n\k | 0 1 2 3 4 5 6 ----+---------------------------------------------------------- 0 | 1 1 1 1 1 1 1 ... 1 | 1 1 3 13 75 541 4683 ... 2 | 1 0 6 120 6174 449520 49686726 ... 3 | 1 0 0 1104 413088 329520720 491236986720 ... 4 | 1 0 0 5040 18481080 179438982360 3785623968170400 ... 5 | 1 0 0 0 522481680 70302503250720 ... 6 | 1 0 0 0 7875584640 ... ... The A(2,2) = 6 matrices are: [1 1] [1 1] [1 0] [1 0] [0 1] [0 1] [1 0] [0 1] [1 1] [0 1] [1 1] [1 0] [0 1] [1 0] [0 1] [1 1] [1 0] [1 1]
Links
- Andrew Howroyd, Table of n, a(n) for n = 0..209
Crossrefs
Programs
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PARI
WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)} D(p, n, k)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); WeighT(v)[n]^k/prod(i=1, #v, i^v[i]*v[i]!)} 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]))); }
Formula
A(n,k) = 0 for k > 0, n > 2^(k-1).
A(2^(k-1), k) = (2^k-1)! for k > 0.
A331643(n) = Sum_{d|n} A(n/d, d).