A318795 Array read by antidiagonals: T(n,k) is the number of inequivalent nonnegative integer n X n matrices with sum of elements equal to k, under row and column permutations.
1, 1, 1, 1, 4, 1, 1, 5, 4, 1, 1, 11, 10, 4, 1, 1, 14, 24, 10, 4, 1, 1, 24, 51, 33, 10, 4, 1, 1, 30, 114, 78, 33, 10, 4, 1, 1, 45, 219, 224, 91, 33, 10, 4, 1, 1, 55, 424, 549, 277, 91, 33, 10, 4, 1, 1, 76, 768, 1403, 792, 298, 91, 33, 10, 4, 1, 1, 91, 1352, 3292, 2341, 881, 298, 91, 33, 10, 4, 1
Offset: 1
Examples
Array begins: =========================================================== n\k| 1 2 3 4 5 6 7 8 9 10 11 12 ---+------------------------------------------------------- 1 | 1 1 1 1 1 1 1 1 1 1 1 1 ... 2 | 1 4 5 11 14 24 30 45 55 76 91 119 ... 3 | 1 4 10 24 51 114 219 424 768 1352 2278 3759 ... 4 | 1 4 10 33 78 224 549 1403 3292 7677 16934 36581 ... 5 | 1 4 10 33 91 277 792 2341 6654 18802 51508 138147 ... 6 | 1 4 10 33 91 298 881 2825 8791 27947 87410 272991 ... 7 | 1 4 10 33 91 298 910 2974 9655 32287 108274 367489 ... 8 | 1 4 10 33 91 298 910 3017 9886 33767 116325 410298 ... 9 | 1 4 10 33 91 298 910 3017 9945 34124 118729 424498 ... ...
Links
- Andrew Howroyd, Table of n, a(n) for n = 1..1275
- Andrew Howroyd, Additional PARI Programs
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[1/Product[(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+1, n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Sep 12 2018, after Andrew Howroyd *) -
PARI
\\ see also link. 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} K(q, t, k)={1/prod(j=1, #q, (1-y^lcm(t,q[j]) + O(y*y^k))^gcd(t, q[j]))} M(m, n, k)={my(s=0); forpart(q=m, s+=permcount(q)*polcoef(polcoef(exp(sum(t=1, n, K(q, t, k)/t*x^t) + O(x*x^n)), n), k)); s/m!} for(n=1, 10, for(k=1, 12, print1(M(n, n, k), ", ")); print); \\ updated Andrew Howroyd, Mar 29 2020
Formula
T(n,k) = T(k,k) for n > k.