A330679 -id:A330679 - cp's OEIS frontend

A330726 Number of balanced reduced multisystems of maximum depth whose atoms are positive integers summing to n.

1, 1, 2, 3, 7, 17, 54, 199, 869, 4341, 24514, 154187

Offset: 0

Gus Wiseman, Jan 03 2020

A balanced reduced multisystem is either a finite multiset, or a multiset partition with at least two parts, not all of which are singletons, of a balanced reduced multisystem.

			The a(1) = 1 through a(5) = 17 multisystems (commas elided):
  {1}  {2}   {3}        {4}               {5}
       {11}  {12}       {22}              {23}
             {{1}{11}}  {13}              {14}
                        {{1}{12}}         {{1}{13}}
                        {{2}{11}}         {{1}{22}}
                        {{{1}}{{1}{11}}}  {{2}{12}}
                        {{{11}}{{1}{1}}}  {{3}{11}}
                                          {{{1}}{{1}{12}}}
                                          {{{11}}{{1}{2}}}
                                          {{{1}}{{2}{11}}}
                                          {{{12}}{{1}{1}}}
                                          {{{2}}{{1}{11}}}
                                          {{{{1}}}{{{1}}{{1}{11}}}}
                                          {{{{1}}}{{{11}}{{1}{1}}}}
                                          {{{{1}{1}}}{{{1}}{{11}}}}
                                          {{{{1}{11}}}{{{1}}{{1}}}}
                                          {{{{11}}}{{{1}}{{1}{1}}}}

The case with all atoms equal to 1 is A000111.
The non-maximal version is A330679.
A tree version is A320160.
Cf. A000669, A002846, A005121, A141268, A196545, A213427, A317145, A318813, A330663, A330665, A330675, A330676, A330728.

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]]]];
totm[m_]:=Prepend[Join@@Table[totm[p],{p,Select[mps[m],1

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.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jan 03 2020

			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)

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.

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}]

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

cp's OEIS Frontend

A330726 Number of balanced reduced multisystems of maximum depth whose atoms are positive integers summing to n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica