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A324171 Number of non-crossing multiset partitions of normal multisets of size n. 14
1, 1, 4, 16, 75, 378, 2042, 11489, 66697 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

A multiset is normal if its union is an initial interval of positive integers.

A multiset partition is crossing if it has a 2-element submultiset of the form {{...x...y...}, {...z...t...}} where x < z < y < t or z < x < t < y.

LINKS

Table of n, a(n) for n=0..8.

EXAMPLE

The A255906(5) - a(5) = 22 crossing multiset partitions:

  {{13}{124}}  {{1}{13}{24}}

  {{13}{224}}  {{1}{24}{35}}

  {{13}{234}}  {{2}{13}{24}}

  {{13}{244}}  {{2}{14}{35}}

  {{13}{245}}  {{3}{13}{24}}

  {{14}{235}}  {{3}{14}{25}}

  {{24}{113}}  {{4}{13}{24}}

  {{24}{123}}  {{4}{13}{25}}

  {{24}{133}}  {{5}{13}{24}}

  {{24}{134}}

  {{24}{135}}

  {{25}{134}}

  {{35}{124}}

MATHEMATICA

nonXQ[stn_]:=!MatchQ[stn, {___, {___, x_, ___, y_, ___}, ___, {___, z_, ___, t_, ___}, ___}/; x<z<y<t||z<x<t<y];

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]]]];

allnorm[n_]:=If[n<=0, {{}}, Function[s, Array[Count[s, y_/; y<=#]+1&, n]]/@Subsets[Range[n-1]+1]];

Table[Sum[Length[Select[mps[m], nonXQ]], {m, allnorm[n]}], {n, 0, 8}]

CROSSREFS

Cf. A000108 (non-crossing set partitions), A000124, A001006, A001055, A001263, A007297, A054726 (non-crossing graphs), A099947, A194560, A255906 (multiset partitions of normal multisets), A306438.

Cf. A324166, A324167, A324168, A324169, A324170, A324173.

Sequence in context: A331159 A101205 A301577 * A204772 A050540 A094559

Adjacent sequences:  A324168 A324169 A324170 * A324172 A324173 A324174

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Feb 17 2019

STATUS

approved

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Last modified January 28 14:26 EST 2022. Contains 350656 sequences. (Running on oeis4.)