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

145, 169, 215, 290, 338, 355, 377, 395, 430, 435, 473, 481, 505, 507, 535, 559, 565, 580, 645, 667, 676, 695, 710, 725, 754, 790, 793, 803, 815, 841, 845, 860, 865, 869, 870, 905, 923, 946, 962, 965, 989, 995, 1010, 1014, 1015, 1027, 1065, 1070, 1073, 1075

Offset: 1

Author

Gus Wiseman, Jun 22 2019

Keywords

• nonn

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 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.

Examples

			The sequence of terms together with their multiset multisystems begins:
  145: {{2},{1,3}}
  169: {{1,2},{1,2}}
  215: {{2},{1,4}}
  290: {{},{2},{1,3}}
  338: {{},{1,2},{1,2}}
  355: {{2},{1,1,3}}
  377: {{1,2},{1,3}}
  395: {{2},{1,5}}
  430: {{},{2},{1,4}}
  435: {{1},{2},{1,3}}
  473: {{3},{1,4}}
  481: {{1,2},{1,1,2}}
  505: {{2},{1,6}}
  507: {{1},{1,2},{1,2}}
  535: {{2},{1,1,4}}
  559: {{1,2},{1,4}}
  565: {{2},{1,2,3}}
  580: {{},{},{2},{1,3}}
  645: {{1},{2},{1,4}}
  667: {{2,2},{1,3}}
		

Crossrefs

Unsortable set partitions are A058681.

Normal unsortable multiset partitions are A326211.

Unsortable digraphs are A326209.

MM-numbers of crossing multiset partitions are A324170.

MM-numbers of nesting multiset partitions are A326256.

MM-numbers of capturing multiset partitions are A326255.

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

Cf. A326212, A326243, A326257, A326259.

Programs

Mathematica

lexsort[f_,c_]:=OrderedQ[PadRight[{f,c}]];
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],!OrderedQ[Join@@Sort[primeMS/@primeMS[#],lexsort]]&]