|
%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
|
