A326211 Number of unsortable normal multiset partitions of weight n.

0, 0, 0, 1, 17, 170, 1455, 11678, 92871, 752473, 6316053, 55407937, 509989738, 4928781193, 49964562989, 530288898560, 5879533995923, 67951648386065, 816923847528339, 10196598719427105, 131904798148590231, 1765644786688907017, 24420201210245332196

Offset

0,5

Comments

A multiset partition is normal if it covers an initial interval of positive integers. It is unsortable if no permutation has an ordered concatenation, or equivalently if the concatenation of its lexicographically-ordered parts is not weakly increasing. For example, the multiset partition {{1,2},{1,1,1},{2,2,2}} is sortable because the permutation ((1,1,1),(1,2),(2,2,2)) has concatenation (1,1,1,1,2,2,2,2), which is weakly increasing.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..400

Formula

a(n) = A255906(n) - A326212(n).

Example

The a(3) = 1 and a(4) = 17 multiset partitions:
  {{1,3},{2}}  {{1,1,3},{2}}
               {{1,2},{1,2}}
               {{1,2},{1,3}}
               {{1,2,3},{2}}
               {{1,2,4},{3}}
               {{1,3},{2,2}}
               {{1,3},{2,3}}
               {{1,3},{2,4}}
               {{1,3,3},{2}}
               {{1,3,4},{2}}
               {{1,4},{2,3}}
               {{1},{1,3},{2}}
               {{1},{2,4},{3}}
               {{1,3},{2},{2}}
               {{1,3},{2},{3}}
               {{1,3},{2},{4}}
               {{1,4},{2},{3}}

Mathematica

lexsort[f_, c_]:=OrderedQ[PadRight[{f, c}]];
allnorm[n_]:=If[n<=0, {{}}, Function[s, Array[Count[s, y_/; y<=#]+1&, n]]/@Subsets[Range[n-1]+1]];
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]]]];
Table[Length[Select[Sort[#, lexsort]&/@Join@@mps/@allnorm[n], !OrderedQ[Join@@#]&]], {n, 0, 5}]

Crossrefs

Unsortable set partitions are A058681.

Sortable normal multiset partitions are A326212.

Non-crossing normal multiset partitions are A324171.

MM-numbers of unsortable multiset partitions are A326258.

Cf. A000108, A016098, A255906, A324170.

Cf. A326209, A326210, A326243, A326250, A326255, A326256.

Sequence in context: A282922 A023015 A022645 * A164747 A166579 A142169

Adjacent sequences: A326208 A326209 A326210 * A326212 A326213 A326214

Keyword

nonn

Author

Gus Wiseman, Jun 19 2019

Extensions

a(10) onward from A255906(n)-A326212(n) by Martin Fuller, Feb 27 2026

Status

approved