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A326259 MM-numbers of crossing, capturing multiset partitions (with empty parts allowed). 6
8903, 15167, 16717, 17806, 18647, 20329, 20453, 21797, 22489, 25607, 26709, 27649, 29551, 30334, 31373, 32741, 33434, 34691, 35177, 35612, 35821, 37091, 37133, 37294, 37969, 38243, 39493, 40658, 40906, 41449, 42011, 42949, 43594, 43817, 43873, 44515, 44861 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The multiset multisystem with MM-number n is obtained by taking the multiset of prime indices of each prime index of n.

A multiset partition is crossing if it has two blocks of the form {...x...y...}, {...z...t...} where x < z < y < t or z < x < t < y. It is capturing if it has two blocks of the form {...x...y...} and {...z...t...} where x < z and y > t or x > z and y < t. Capturing is a weaker condition than nesting, so for example {{1,3,5},{2,4}} is capturing but not nesting.

LINKS

Table of n, a(n) for n=1..37.

EXAMPLE

The sequence of terms together with their multiset multisystems begins:

   8903: {{1,3},{2,2,4}}

  15167: {{1,3},{2,2,5}}

  16717: {{2,4},{1,3,3}}

  17806: {{},{1,3},{2,2,4}}

  18647: {{1,3},{2,2,6}}

  20329: {{1,3},{1,2,2,4}}

  20453: {{1,2,3},{1,2,4}}

  21797: {{1,1,3},{2,2,4}}

  22489: {{1,4},{2,2,5}}

  25607: {{1,3},{2,2,7}}

  26709: {{1},{1,3},{2,2,4}}

  27649: {{1,4},{2,2,6}}

  29551: {{1,3},{2,2,8}}

  30334: {{},{1,3},{2,2,5}}

  31373: {{2,5},{1,3,3}}

  32741: {{1,3},{2,2,2,4}}

  33434: {{},{2,4},{1,3,3}}

  34691: {{1,2,3},{2,2,4}}

  35177: {{1,3},{1,2,2,5}}

  35612: {{},{},{1,3},{2,2,4}}

MATHEMATICA

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

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

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];

Select[Range[100000], capXQ[primeMS/@primeMS[#]]&&croXQ[primeMS/@primeMS[#]]&]

CROSSREFS

Crossing set partitions are A000108.

Capturing set partitions are A326243.

Crossing, capturing set partitions are A326246.

MM-numbers of crossing multiset partitions are A324170.

MM-numbers of nesting multiset partitions are A326256.

MM-numbers of capturing multiset partitions are A326255.

MM-numbers of unsortable multiset partitions are A326258.

Cf. A001055, A001519, A016098, A056239, A058681, A112798, A122880, A302242.

Cf. A326211, A326245, A326248, A326249, A326254.

Sequence in context: A253664 A037192 A236890 * A206235 A329799 A329468

Adjacent sequences:  A326256 A326257 A326258 * A326260 A326261 A326262

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jun 22 2019

STATUS

approved

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Last modified August 4 08:57 EDT 2021. Contains 346445 sequences. (Running on oeis4.)