A326258 - OEIS

%I #5 Jun 22 2019 23:14:11

%S 145,169,215,290,338,355,377,395,430,435,473,481,505,507,535,559,565,

%T 580,645,667,676,695,710,725,754,790,793,803,815,841,845,860,865,869,

%U 870,905,923,946,962,965,989,995,1010,1014,1015,1027,1065,1070,1073,1075

%N MM-numbers of unsortable multiset partitions (with empty parts allowed).

%C 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.

%C A multiset partition is unsortable if no permutation has an ordered concatenation. 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.

%e The sequence of terms together with their multiset multisystems begins:

%e   145: {{2},{1,3}}

%e   169: {{1,2},{1,2}}

%e   215: {{2},{1,4}}

%e   290: {{},{2},{1,3}}

%e   338: {{},{1,2},{1,2}}

%e   355: {{2},{1,1,3}}

%e   377: {{1,2},{1,3}}

%e   395: {{2},{1,5}}

%e   430: {{},{2},{1,4}}

%e   435: {{1},{2},{1,3}}

%e   473: {{3},{1,4}}

%e   481: {{1,2},{1,1,2}}

%e   505: {{2},{1,6}}

%e   507: {{1},{1,2},{1,2}}

%e   535: {{2},{1,1,4}}

%e   559: {{1,2},{1,4}}

%e   565: {{2},{1,2,3}}

%e   580: {{},{},{2},{1,3}}

%e   645: {{1},{2},{1,4}}

%e   667: {{2,2},{1,3}}

%t lexsort[f_,c_]:=OrderedQ[PadRight[{f,c}]];

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

%t Select[Range[1000],!OrderedQ[Join@@Sort[primeMS/@primeMS[#],lexsort]]&]

%Y Unsortable set partitions are A058681.

%Y Normal unsortable multiset partitions are A326211.

%Y Unsortable digraphs are A326209.

%Y MM-numbers of crossing multiset partitions are A324170.

%Y MM-numbers of nesting multiset partitions are A326256.

%Y MM-numbers of capturing multiset partitions are A326255.

%Y Cf. A001055, A016098, A056239, A112798, A302242.

%Y Cf. A326212, A326243, A326257, A326259.

%K nonn

%O 1,1

%A _Gus Wiseman_, Jun 22 2019