cp's OEIS Frontend

A386579 Number of permutations of row n of A305936 (a multiset whose multiplicities are the prime indices of n) with k adjacent unequal parts.

%I A386579 #5 Aug 05 2025 21:33:19
%S A386579 1,1,0,0,2,1,0,0,0,2,1,1,0,0,0,0,0,6,0,2,2,2,0,2,2,0,1,0,0,0,0,0,0,6,
%T A386579 6,1,0,0,0,0,0,0,2,3,0,0,0,2,3,4,1,0,0,0,24,1,0,0,0,0,0,0,0,0,6,12,12,
%U A386579 1,0,0,0,0,0,0,0,0,0,6,12,2,0,2,4,6,3,0
%N A386579 Number of permutations of row n of A305936 (a multiset whose multiplicities are the prime indices of n) with k adjacent unequal parts.
%C A386579 Row 1 is empty, so offset is 2.
%C A386579 Same as A386578 with rows reversed.
%C A386579 This multiset (row n of A305936) is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.
%e A386579 Row n = 21 counts the following permutations:
%e A386579   .  111122  111221  111212  112121  .
%e A386579      221111  112211  112112  121121
%e A386579              122111  121112  121211
%e A386579              211112  211121
%e A386579                      211211
%e A386579                      212111
%e A386579 Triangle begins:
%e A386579   .
%e A386579   1
%e A386579   1  0
%e A386579   0  2
%e A386579   1  0  0
%e A386579   0  2  1
%e A386579   1  0  0  0
%e A386579   0  0  6
%e A386579   0  2  2  2
%e A386579   0  2  2  0
%e A386579   1  0  0  0  0
%e A386579   0  0  6  6
%e A386579   1  0  0  0  0  0
%e A386579   0  2  3  0  0
%e A386579   0  2  3  4  1
%e A386579   0  0  0 24
%e A386579   1  0  0  0  0  0  0
%e A386579   0  0  6 12 12
%e A386579   1  0  0  0  0  0  0  0
%e A386579   0  0  6 12  2
%e A386579   0  2  4  6  3  0
%t A386579 nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
%t A386579 ugt[c_,x_]:=Select[Permutations[c],Function[q,Length[Select[Range[Length[q]-1],q[[#]]!=q[[#+1]]&]]==x]];
%t A386579 Table[Table[Length[ugt[nrmptn[n],k]],{k,0,Length[nrmptn[n]]-1}],{n,30}]
%Y A386579 Column k = 0 is A010051.
%Y A386579 Row lengths are A056239.
%Y A386579 Row sums are A318762.
%Y A386579 Column k = last is A335125.
%Y A386579 For prime indices we have A374252, reverse A386577.
%Y A386579 Reversing all rows gives A386578.
%Y A386579 A003242 and A335452 count anti-runs, ranks A333489, patterns A005649.
%Y A386579 A025065(n - 2) counts partitions of inseparable type, ranks A335126, sums of A386586.
%Y A386579 A124762 gives inseparability of standard compositions, separability A333382.
%Y A386579 A305936 is a multiset whose multiplicities are the prime indices of n.
%Y A386579 A325534 counts separable multisets, ranks A335433, sums of A386583.
%Y A386579 A325535 counts inseparable multisets, ranks A335448, sums of A386584.
%Y A386579 A336106 counts partitions of separable type, ranks A335127, sums of A386585.
%Y A386579 Cf. A001221, A001222, A003963, A051903, A051904, A106351, A112798, A238130, A336102, A373957, A386581.
%K A386579 nonn,tabf
%O A386579 2,5
%A A386579 _Gus Wiseman_, Aug 04 2025