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A326258 MM-numbers of unsortable multiset partitions (with empty parts allowed).

%I A326258 #5 Jun 22 2019 23:14:11
%S A326258 145,169,215,290,338,355,377,395,430,435,473,481,505,507,535,559,565,
%T A326258 580,645,667,676,695,710,725,754,790,793,803,815,841,845,860,865,869,
%U A326258 870,905,923,946,962,965,989,995,1010,1014,1015,1027,1065,1070,1073,1075
%N A326258 MM-numbers of unsortable multiset partitions (with empty parts allowed).
%C A326258 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 A326258 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 A326258 The sequence of terms together with their multiset multisystems begins:
%e A326258   145: {{2},{1,3}}
%e A326258   169: {{1,2},{1,2}}
%e A326258   215: {{2},{1,4}}
%e A326258   290: {{},{2},{1,3}}
%e A326258   338: {{},{1,2},{1,2}}
%e A326258   355: {{2},{1,1,3}}
%e A326258   377: {{1,2},{1,3}}
%e A326258   395: {{2},{1,5}}
%e A326258   430: {{},{2},{1,4}}
%e A326258   435: {{1},{2},{1,3}}
%e A326258   473: {{3},{1,4}}
%e A326258   481: {{1,2},{1,1,2}}
%e A326258   505: {{2},{1,6}}
%e A326258   507: {{1},{1,2},{1,2}}
%e A326258   535: {{2},{1,1,4}}
%e A326258   559: {{1,2},{1,4}}
%e A326258   565: {{2},{1,2,3}}
%e A326258   580: {{},{},{2},{1,3}}
%e A326258   645: {{1},{2},{1,4}}
%e A326258   667: {{2,2},{1,3}}
%t A326258 lexsort[f_,c_]:=OrderedQ[PadRight[{f,c}]];
%t A326258 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A326258 Select[Range[1000],!OrderedQ[Join@@Sort[primeMS/@primeMS[#],lexsort]]&]
%Y A326258 Unsortable set partitions are A058681.
%Y A326258 Normal unsortable multiset partitions are A326211.
%Y A326258 Unsortable digraphs are A326209.
%Y A326258 MM-numbers of crossing multiset partitions are A324170.
%Y A326258 MM-numbers of nesting multiset partitions are A326256.
%Y A326258 MM-numbers of capturing multiset partitions are A326255.
%Y A326258 Cf. A001055, A016098, A056239, A112798, A302242.
%Y A326258 Cf. A326212, A326243, A326257, A326259.
%K A326258 nonn
%O A326258 1,1
%A A326258 _Gus Wiseman_, Jun 22 2019