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A318795
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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.
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15
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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
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OFFSET
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1,5
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LINKS
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Andrew Howroyd, Table of n, a(n) for n = 1..1275
Andrew Howroyd, Additional PARI Programs
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FORMULA
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T(n,k) = T(k,k) for n > k.
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EXAMPLE
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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 ...
...
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Rows 2..7 are A053307, A052365, A052366, A052367, A052372, A052373.
Main diagonal is A007716.
Cf. A214398, A246106, A304942, A318805.
Sequence in context: A021247 A016522 A153843 * A099575 A173740 A028275
Adjacent sequences: A318792 A318793 A318794 * A318796 A318797 A318798
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KEYWORD
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nonn,tabl
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AUTHOR
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Andrew Howroyd, Sep 03 2018
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STATUS
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approved
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