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A317533
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Regular triangle read rows: T(n,k) = number of non-isomorphic multiset partitions of size n and length k.
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73
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1, 2, 2, 3, 4, 3, 5, 14, 9, 5, 7, 28, 33, 16, 7, 11, 69, 104, 74, 29, 11, 15, 134, 294, 263, 142, 47, 15, 22, 285, 801, 948, 599, 263, 77, 22, 30, 536, 2081, 3058, 2425, 1214, 453, 118, 30, 42, 1050, 5212, 9769, 9276, 5552, 2322, 761, 181, 42, 56, 1918, 12645, 29538, 34172, 23770, 11545, 4179, 1223, 267, 56
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OFFSET
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1,2
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LINKS
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EXAMPLE
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Non-isomorphic representatives of the T(3,2) = 4 multiset partitions:
{{1},{1,1}}
{{1},{1,2}}
{{1},{2,2}}
{{1},{2,3}}
Triangle begins:
1
2 2
3 4 3
5 14 9 5
7 28 33 16 7
11 69 104 74 29 11
15 134 294 263 142 47 15
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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!)];
T[n_, k_] := M[k, n, n] - M[k - 1, n, n];
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PROG
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(PARI) \\ See A318795 for definition of M.
T(n, k)={M(k, n, n) - M(k-1, n, n)}
for(n=1, 10, for(k=1, n, print1(T(n, k), ", ")); print) \\ Andrew Howroyd, Dec 28 2019
(PARI) \\ Faster version.
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, n)={1/prod(j=1, #q, (1-x^lcm(t, q[j]) + O(x*x^n))^gcd(t, q[j]))}
G(m, n)={my(s=0); forpart(q=m, s+=permcount(q)*exp(sum(t=1, n, (K(q, t, n)-1)/t) + O(x*x^n))); s/m!}
A(n, m=n)={my(p=sum(k=0, m, G(k, n)*y^k)*(1-y)); matrix(n, m, n, k, polcoef(polcoef(p, n, x), k, y))}
{ my(T=A(10)); for(n=1, #T, print(T[n, 1..n])) } \\ Andrew Howroyd, Aug 30 2020
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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