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A322133
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Regular triangle read by rows where T(n,k) is the number of non-isomorphic connected multiset partitions of weight n with k vertices.
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4
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1, 0, 1, 0, 2, 1, 0, 3, 2, 1, 0, 5, 8, 3, 1, 0, 7, 17, 12, 3, 1, 0, 11, 46, 45, 18, 4, 1, 0, 15, 94, 141, 76, 23, 4, 1, 0, 22, 212, 432, 333, 124, 30, 5, 1, 0, 30, 416, 1231, 1254, 622, 178, 37, 5, 1, 0, 42, 848, 3346, 4601, 2914, 1058, 252, 45, 6, 1
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OFFSET
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0,5
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COMMENTS
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The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.
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LINKS
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EXAMPLE
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Triangle begins:
1
0 1
0 2 1
0 3 2 1
0 5 8 3 1
0 7 17 12 3 1
0 11 46 45 18 4 1
0 15 94 141 76 23 4 1
0 22 212 432 333 124 30 5 1
0 30 416 1231 1254 622 178 37 5 1
0 42 848 3346 4601 2914 1058 252 45 6 1
Non-isomorphic representatives of the multiset partitions counted in row 4:
{{1,1,1,1}} {{1,1,2,2}} {{1,2,3,3}} {{1,2,3,4}}
{{1},{1,1,1}} {{1,2,2,2}} {{1,3},{2,3}}
{{1,1},{1,1}} {{1},{1,2,2}} {{3},{1,2,3}}
{{1},{1},{1,1}} {{1,2},{1,2}}
{{1},{1},{1},{1}} {{1,2},{2,2}}
{{2},{1,2,2}}
{{1},{2},{1,2}}
{{2},{2},{1,2}}
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PROG
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(PARI) \\ Needs G(m, n) defined in A317533 (faster PARI).
InvEulerMTS(p)={my(n=serprec(p, x)-1, q=log(p), vars=variables(p)); sum(i=1, n, moebius(i)*substvec(q + O(x*x^(n\i)), vars, apply(v->v^i, vars))/i)}
T(n)={[Vecrev(p) | p <- Vec(1 + InvEulerMTS(y^n*G(n, n) + sum(k=0, n-1, y^k*(1 - y)*G(k, n))))]}
{ my(A=T(10)); for(i=1, #A, print(A[i])) } \\ Andrew Howroyd, Jan 15 2024
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CROSSREFS
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Cf. A000664, A007716, A054923, A191646, A191970, A275421, A286520, A317672, A319719, A321155, A321254, A322114.
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KEYWORD
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AUTHOR
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STATUS
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approved
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