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A330785
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Triangle read by rows where T(n,k) is the number of chains of length k from minimum to maximum in the poset of integer partitions of n ordered by refinement.
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1
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1, 0, 1, 0, 1, 1, 0, 1, 3, 2, 0, 1, 5, 8, 4, 0, 1, 9, 25, 28, 11, 0, 1, 13, 57, 111, 99, 33, 0, 1, 20, 129, 379, 561, 408, 116, 0, 1, 28, 253, 1057, 2332, 2805, 1739, 435, 0, 1, 40, 496, 2833, 8695, 15271, 15373, 8253, 1832, 0, 1, 54, 898, 6824, 28071, 67790, 98946, 85870, 40789, 8167
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OFFSET
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1,9
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LINKS
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FORMULA
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EXAMPLE
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Triangle begins:
1
0 1
0 1 1
0 1 3 2
0 1 5 8 4
0 1 9 25 28 11
0 1 13 57 111 99 33
0 1 20 129 379 561 408 116
Row n = 5 counts the following chains (minimum and maximum not shown):
() (14) (113)->(14) (1112)->(113)->(14)
(23) (113)->(23) (1112)->(113)->(23)
(113) (122)->(14) (1112)->(122)->(14)
(122) (122)->(23) (1112)->(122)->(23)
(1112) (1112)->(14)
(1112)->(23)
(1112)->(113)
(1112)->(122)
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MATHEMATICA
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sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
upr[q_]:=Union[Sort/@Apply[Plus, mps[q], {2}]];
paths[eds_, start_, end_]:=If[start==end, Prepend[#, {}], #]&[Join@@Table[Prepend[#, e]&/@paths[eds, Last[e], end], {e, Select[eds, First[#]==start&]}]];
Table[Length[Select[paths[Join@@Table[{y, #}&/@DeleteCases[upr[y], y], {y, Sort/@IntegerPartitions[n]}], ConstantArray[1, n], {n}], Length[#]==k-1&]], {n, 8}, {k, n}]
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CROSSREFS
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The version for set partitions is A008826.
The version for factorizations is A330935.
Cf. A000111, A000258, A000311, A005121, A141268, A196545, A265947, A300383, A306186, A317141, A317176, A318813, A320160, A330679.
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KEYWORD
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AUTHOR
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STATUS
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approved
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