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 A330785 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. 1
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,9 LINKS FORMULA T(n,k) = A330935(2^n,k). EXAMPLE 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) MATHEMATICA 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}] CROSSREFS Row sums are A213427. Main diagonal is A002846. Column k=3 is A007042. Dominated by A330784. 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. Sequence in context: A008783 A139144 A081576 * A292717 A054654 A253669 Adjacent sequences:  A330782 A330783 A330784 * A330786 A330787 A330788 KEYWORD nonn,tabl AUTHOR Gus Wiseman, Jan 03 2020 STATUS approved

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Last modified April 10 11:15 EDT 2021. Contains 342845 sequences. (Running on oeis4.)