A318795 - OEIS

%I #14 Mar 29 2020 15:47:53

%S 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,

%T 30,114,78,33,10,4,1,1,45,219,224,91,33,10,4,1,1,55,424,549,277,91,33,

%U 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

%N 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.

%H Andrew Howroyd, <a href="/A318795/b318795.txt">Table of n, a(n) for n = 1..1275</a>

%H Andrew Howroyd, <a href="/A318795/a318795.txt">Additional PARI Programs</a>

%F T(n,k) = T(k,k) for n > k.

%e Array begins:

%e ===========================================================

%e n\k| 1 2  3  4  5   6   7    8    9    10     11     12

%e ---+-------------------------------------------------------

%e 1  | 1 1  1  1  1   1   1    1    1     1      1      1 ...

%e 2  | 1 4  5 11 14  24  30   45   55    76     91    119 ...

%e 3  | 1 4 10 24 51 114 219  424  768  1352   2278   3759 ...

%e 4  | 1 4 10 33 78 224 549 1403 3292  7677  16934  36581 ...

%e 5  | 1 4 10 33 91 277 792 2341 6654 18802  51508 138147 ...

%e 6  | 1 4 10 33 91 298 881 2825 8791 27947  87410 272991 ...

%e 7  | 1 4 10 33 91 298 910 2974 9655 32287 108274 367489 ...

%e 8  | 1 4 10 33 91 298 910 3017 9886 33767 116325 410298 ...

%e 9  | 1 4 10 33 91 298 910 3017 9945 34124 118729 424498 ...

%e ...

%t 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];

%t 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}];

%t 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!)];

%t 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_ *)

%o (PARI) \\ see also link.

%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)={1/prod(j=1, #q, (1-y^lcm(t,q[j]) + O(y*y^k))^gcd(t, q[j]))}

%o 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!}

%o for(n=1, 10, for(k=1, 12, print1(M(n, n, k), ", ")); print); \\ updated _Andrew Howroyd_, Mar 29 2020

%Y Rows 2..7 are A053307, A052365, A052366, A052367, A052372, A052373.

%Y Main diagonal is A007716.

%Y Cf. A214398, A246106, A304942, A318805.

%K nonn,tabl

%O 1,5

%A _Andrew Howroyd_, Sep 03 2018