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A326259
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MM-numbers of crossing, capturing multiset partitions (with empty parts allowed).
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6
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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
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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}}
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MATHEMATICA
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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[#]]&]
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CROSSREFS
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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.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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