A321609 Array read by antidiagonals: T(n,k) is the number of inequivalent binary n X n matrices with k ones, under row and column permutations.
1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 3, 1, 1, 0, 0, 1, 3, 1, 1, 0, 0, 1, 6, 3, 1, 1, 0, 0, 0, 7, 6, 3, 1, 1, 0, 0, 0, 7, 16, 6, 3, 1, 1, 0, 0, 0, 6, 21, 16, 6, 3, 1, 1, 0, 0, 0, 3, 39, 34, 16, 6, 3, 1, 1, 0, 0, 0, 1, 44, 69, 34, 16, 6, 3, 1, 1, 0, 0, 0, 1, 55, 130, 90, 34, 16, 6, 3, 1, 1
Offset: 0
Examples
Array begins: ========================================================== n\k| 0 1 2 3 4 5 6 7 8 9 10 11 12 ---+------------------------------------------------------ 0 | 1 0 0 0 0 0 0 0 0 0 0 0 0 ... 1 | 1 1 0 0 0 0 0 0 0 0 0 0 0 ... 2 | 1 1 3 1 1 0 0 0 0 0 0 0 0 ... 3 | 1 1 3 6 7 7 6 3 1 1 0 0 0 ... 4 | 1 1 3 6 16 21 39 44 55 44 39 21 16 ... 5 | 1 1 3 6 16 34 69 130 234 367 527 669 755 ... 6 | 1 1 3 6 16 34 90 182 425 870 1799 3323 5973 ... 7 | 1 1 3 6 16 34 90 211 515 1229 2960 6893 15753 ... 8 | 1 1 3 6 16 34 90 211 558 1371 3601 9209 24110 ... 9 | 1 1 3 6 16 34 90 211 558 1430 3825 10278 28427 ... ...
Crossrefs
Programs
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Mathematica
permcount[v_List] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t*k; s += t]; s!/m]; c[p_List, q_List, k_] := SeriesCoefficient[Product[Product[(1 + O[x]^(k + 1) + x^LCM[p[[i]], q[[j]]])^GCD[p[[i]], q[[j]]], {j, 1, Length[q]}], {i, 1, Length[p]}], {x, 0, k}]; M[m_, n_, k_] := Module[{s = 0}, Do[Do[s += permcount[p]*permcount[q]*c[p, q, k], {q, IntegerPartitions[n]}], {p, IntegerPartitions[m]}]; s/(m!*n!)] Table[M[n - k, n - k, k], {n, 0, 12}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Sep 10 2019, after Andrew Howroyd *) -
PARI
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} c(p, q, k)={polcoef(prod(i=1, #p, prod(j=1, #q, (1 + x^lcm(p[i], q[j]) + O(x*x^k))^gcd(p[i], q[j]))), k)} M(m, n, k)={my(s=0); forpart(p=m, forpart(q=n, s+=permcount(p) * permcount(q) * c(p, q, k))); s/(m!*n!)} for(n=0, 10, for(k=0, 12, print1(M(n, n, k), ", ")); print); \\ Andrew Howroyd, Nov 14 2018
Formula
T(n,k) = T(k,k) for n > k.
T(n,k) = 0 for k > n^2.