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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.

%I A330785 #8 Jan 17 2020 14:32:00
%S A330785 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,
%T A330785 20,129,379,561,408,116,0,1,28,253,1057,2332,2805,1739,435,0,1,40,496,
%U A330785 2833,8695,15271,15373,8253,1832,0,1,54,898,6824,28071,67790,98946,85870,40789,8167
%N 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.
%F A330785 T(n,k) = A330935(2^n,k).
%e A330785 Triangle begins:
%e A330785    1
%e A330785    0   1
%e A330785    0   1   1
%e A330785    0   1   3   2
%e A330785    0   1   5   8   4
%e A330785    0   1   9  25  28  11
%e A330785    0   1  13  57 111  99  33
%e A330785    0   1  20 129 379 561 408 116
%e A330785 Row n = 5 counts the following chains (minimum and maximum not shown):
%e A330785   ()  (14)    (113)->(14)    (1112)->(113)->(14)
%e A330785       (23)    (113)->(23)    (1112)->(113)->(23)
%e A330785       (113)   (122)->(14)    (1112)->(122)->(14)
%e A330785       (122)   (122)->(23)    (1112)->(122)->(23)
%e A330785       (1112)  (1112)->(14)
%e A330785               (1112)->(23)
%e A330785               (1112)->(113)
%e A330785               (1112)->(122)
%t A330785 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A330785 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A330785 upr[q_]:=Union[Sort/@Apply[Plus,mps[q],{2}]];
%t A330785 paths[eds_,start_,end_]:=If[start==end,Prepend[#,{}],#]&[Join@@Table[Prepend[#,e]&/@paths[eds,Last[e],end],{e,Select[eds,First[#]==start&]}]];
%t A330785 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}]
%Y A330785 Row sums are A213427.
%Y A330785 Main diagonal is A002846.
%Y A330785 Column k=3 is A007042.
%Y A330785 Dominated by A330784.
%Y A330785 The version for set partitions is A008826.
%Y A330785 The version for factorizations is A330935.
%Y A330785 Cf. A000111, A000258, A000311, A005121, A141268, A196545, A265947, A300383, A306186, A317141, A317176, A318813, A320160, A330679.
%K A330785 nonn,tabl
%O A330785 1,9
%A A330785 _Gus Wiseman_, Jan 03 2020