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A386578 Irregular triangle read by rows where T(n,k) is the number of permutations of row n of A305936 (a multiset whose multiplicities are the prime indices of n) with k adjacent equal parts.

%I A386578 #6 Aug 04 2025 18:41:57
%S A386578 1,0,1,2,0,0,0,1,1,2,0,0,0,0,1,6,0,0,2,2,2,0,0,2,2,0,0,0,0,0,1,6,6,0,
%T A386578 0,0,0,0,0,0,1,0,0,3,2,0,1,4,3,2,0,24,0,0,0,0,0,0,0,0,0,1,12,12,6,0,0,
%U A386578 0,0,0,0,0,0,0,1,2,12,6,0,0,0,3,6,4,2,0
%N A386578 Irregular triangle read by rows where T(n,k) is the number of permutations of row n of A305936 (a multiset whose multiplicities are the prime indices of n) with k adjacent equal parts.
%C A386578 Row 1 is empty, so offset is 2.
%C A386578 Same as A386579 with rows reversed.
%C A386578 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 A386578 Row n = 21 counts the following permutations:
%e A386578   .  112121  111212  111221  111122  .
%e A386578      121121  112112  112211  221111
%e A386578      121211  121112  122111
%e A386578              211121  211112
%e A386578              211211
%e A386578              212111
%e A386578 Triangle begins
%e A386578    .
%e A386578    1
%e A386578    0  1
%e A386578    2  0
%e A386578    0  0  1
%e A386578    1  2  0
%e A386578    0  0  0  1
%e A386578    6  0  0
%e A386578    2  2  2  0
%e A386578    0  2  2  0
%e A386578    0  0  0  0  1
%e A386578    6  6  0  0
%e A386578    0  0  0  0  0  1
%e A386578    0  0  3  2  0
%e A386578    1  4  3  2  0
%e A386578   24  0  0  0
%e A386578    0  0  0  0  0  0  1
%e A386578   12 12  6  0  0
%e A386578    0  0  0  0  0  0  0  1
%e A386578    2 12  6  0  0
%e A386578    0  3  6  4  2  0
%t A386578 nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
%t A386578 aqt[c_,x_]:=Select[Permutations[c],Function[q,Length[Select[Range[Length[q]-1],q[[#]]==q[[#+1]]&]]==x]];
%t A386578 Table[Table[Length[aqt[nrmptn[n],k]],{k,0,Length[nrmptn[n]]-1}],{n,30}]
%Y A386578 Column k = last is A010051.
%Y A386578 Row lengths are A056239.
%Y A386578 Initial zeros are counted by A252736 = A001222 - 1.
%Y A386578 Row sums are A318762.
%Y A386578 Column k = 0 is A335125.
%Y A386578 For prime indices we have A386577.
%Y A386578 Reversing all rows gives A386579.
%Y A386578 A003242 and A335452 count anti-runs, ranks A333489, patterns A005649.
%Y A386578 A025065(n - 2) counts partitions of inseparable type, ranks A335126, sums of A386586.
%Y A386578 A124762 gives inseparability of standard compositions, separability A333382.
%Y A386578 A305936 is a multiset whose multiplicities are the prime indices of n.
%Y A386578 A325534 counts separable multisets, ranks A335433, sums of A386583.
%Y A386578 A325535 counts inseparable multisets, ranks A335448, sums of A386584.
%Y A386578 A336106 counts partitions of separable type, ranks A335127, sums of A386585.
%Y A386578 Cf. A001221, A003963, A051903, A051904, A106351, A112798, A238130, A336102, A373957, A374246, A382525, A386581.
%K A386578 nonn,tabf
%O A386578 2,4
%A A386578 _Gus Wiseman_, Aug 04 2025