A330785 - OEIS

%I #8 Jan 17 2020 14:32:00

%S 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 20,129,379,561,408,116,0,1,28,253,1057,2332,2805,1739,435,0,1,40,496,

%U 2833,8695,15271,15373,8253,1832,0,1,54,898,6824,28071,67790,98946,85870,40789,8167

%N 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 T(n,k) = A330935(2^n,k).

%e Triangle begins:

%e    1

%e    0   1

%e    0   1   1

%e    0   1   3   2

%e    0   1   5   8   4

%e    0   1   9  25  28  11

%e    0   1  13  57 111  99  33

%e    0   1  20 129 379 561 408 116

%e Row n = 5 counts the following chains (minimum and maximum not shown):

%e   ()  (14)    (113)->(14)    (1112)->(113)->(14)

%e       (23)    (113)->(23)    (1112)->(113)->(23)

%e       (113)   (122)->(14)    (1112)->(122)->(14)

%e       (122)   (122)->(23)    (1112)->(122)->(23)

%e       (1112)  (1112)->(14)

%e               (1112)->(23)

%e               (1112)->(113)

%e               (1112)->(122)

%t sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];

%t mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];

%t upr[q_]:=Union[Sort/@Apply[Plus,mps[q],{2}]];

%t paths[eds_,start_,end_]:=If[start==end,Prepend[#,{}],#]&[Join@@Table[Prepend[#,e]&/@paths[eds,Last[e],end],{e,Select[eds,First[#]==start&]}]];

%t 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 Row sums are A213427.

%Y Main diagonal is A002846.

%Y Column k=3 is A007042.

%Y Dominated by A330784.

%Y The version for set partitions is A008826.

%Y The version for factorizations is A330935.

%Y Cf. A000111, A000258, A000311, A005121, A141268, A196545, A265947, A300383, A306186, A317141, A317176, A318813, A320160, A330679.

%K nonn,tabl

%O 1,9

%A _Gus Wiseman_, Jan 03 2020