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A326260
MM-numbers of capturing, non-nesting multiset partitions (with empty parts allowed).
3
2599, 4163, 5198, 6463, 6893, 7291, 7797, 8326, 8507, 9131, 9959, 10396, 10649, 11041, 11639, 12489, 12811, 12926, 12995, 13786, 14237, 14582, 14899, 15157, 15594, 16123, 16403, 16652, 17014, 17063, 17089, 17141, 18101, 18193, 18262, 18643, 18659, 19337, 19389
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 set partition 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. It is nesting 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.
EXAMPLE
The sequence of terms together with their multiset multisystems begins:
2599: {{2,2},{1,2,3}}
4163: {{2,2},{1,2,4}}
5198: {{},{2,2},{1,2,3}}
6463: {{2,2},{1,1,2,3}}
6893: {{1,2,2},{1,2,3}}
7291: {{2,2},{1,2,5}}
7797: {{1},{2,2},{1,2,3}}
8326: {{},{2,2},{1,2,4}}
8507: {{2,3},{1,2,4}}
9131: {{2,2},{1,2,6}}
9959: {{2,2},{1,1,2,4}}
10396: {{},{},{2,2},{1,2,3}}
10649: {{2,2},{1,2,2,3}}
11041: {{1,2,2},{1,2,4}}
11639: {{2,2,2},{1,2,3}}
12489: {{1},{2,2},{1,2,4}}
12811: {{2,2},{1,2,7}}
12926: {{},{2,2},{1,1,2,3}}
12995: {{2},{2,2},{1,2,3}}
13786: {{},{1,2,2},{1,2,3}}
MATHEMATICA
capXQ[stn_]:=MatchQ[stn, {___, {___, x_, ___, y_, ___}, ___, {___, z_, ___, t_, ___}, ___}/; x<z&&t<y||z<x&&y<t];
nesXQ[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[10000], !nesXQ[primeMS/@primeMS[#]]&&capXQ[primeMS/@primeMS[#]]&]
CROSSREFS
Non-nesting set partitions are A000108.
Capturing set partitions are A326243.
Capturing, non-nesting set partitions are A326249.
MM-numbers of nesting multiset partitions are A326256.
MM-numbers of capturing multiset partitions are A326255.
Sequence in context: A236598 A237163 A203257 * A262799 A183965 A031549
KEYWORD
nonn,changed
AUTHOR
Gus Wiseman, Jun 22 2019
STATUS
approved