cp's OEIS Frontend

A386577 Irregular triangle read by rows where T(n,k) is the number of permutations of the multiset of prime factors of n with k adjacent equal terms.

%I A386577 #9 Aug 03 2025 00:05:26
%S A386577 1,1,0,1,1,2,0,1,0,0,1,0,1,2,0,1,1,2,0,1,2,0,2,0,0,0,0,1,1,1,2,0,1,1,
%T A386577 2,0,2,0,2,0,1,0,2,2,0,0,1,2,0,0,0,1,1,2,0,1,6,0,0,1,0,0,0,0,1,2,0,2,
%U A386577 0,2,0,2,2,2,0,1,2,0,2,0,0,2,2,0,1
%N A386577 Irregular triangle read by rows where T(n,k) is the number of permutations of the multiset of prime factors of n with k adjacent equal terms.
%C A386577 Are the rows all unimodal?
%C A386577 Counts permutations of prime factors by "inseparability". For "separability" we have A374252.
%e A386577 The prime indices of 12 are {1,1,2}, and we have:
%e A386577 - 1 permutation (1,2,1) with 0 adjacent equal parts
%e A386577 - 2 permutations (1,1,2), (2,1,1) with 1 adjacent equal part
%e A386577 - 0 permutations with 2 adjacent equal parts
%e A386577 so row 12 is (1,2,0).
%e A386577 Row 48 counts the following permutations:
%e A386577   .  .  (1,1,1,2,1)  (1,1,1,1,2)  .
%e A386577         (1,1,2,1,1)  (2,1,1,1,1)
%e A386577         (1,2,1,1,1)
%e A386577 Row 144 counts the following permutations:
%e A386577   .  (1,1,2,1,2,1)  (1,1,1,2,1,2)  (1,1,1,2,2,1)  (1,1,1,1,2,2)  .
%e A386577      (1,2,1,1,2,1)  (1,1,2,1,1,2)  (1,1,2,2,1,1)  (2,2,1,1,1,1)
%e A386577      (1,2,1,2,1,1)  (1,2,1,1,1,2)  (1,2,2,1,1,1)
%e A386577                     (2,1,1,1,2,1)  (2,1,1,1,1,2)
%e A386577                     (2,1,1,2,1,1)
%e A386577                     (2,1,2,1,1,1)
%e A386577 Triangle begins:
%e A386577    1:
%e A386577    2: 1
%e A386577    3: 1
%e A386577    4: 0  1
%e A386577    6: 1
%e A386577    6: 2  0
%e A386577    7: 1
%e A386577    8: 0  0  1
%e A386577    9: 0  1
%e A386577   10: 2  0
%e A386577   11: 1
%e A386577   12: 1  2  0
%e A386577   13: 1
%e A386577   14: 2  0
%e A386577   15: 2  0
%e A386577   16: 0  0  0  1
%e A386577   17: 1
%e A386577   18: 1  2  0
%e A386577   19: 1
%e A386577   20: 1  2  0
%e A386577   21: 2  0
%e A386577   22: 2  0
%e A386577   23: 1
%e A386577   24: 0  2  2  0
%t A386577 Table[Length[Select[Permutations[Flatten[ConstantArray@@@FactorInteger[n]]],Function[q,Length[Select[Range[Length[q]-1],q[[#]]==q[[#+1]]&]]==k]]],{n,30},{k,0,PrimeOmega[n]-1}]
%Y A386577 Row lengths are A001222.
%Y A386577 The minima of each row are A010051.
%Y A386577 Sorted positions of first appearances appear to be A025487.
%Y A386577 Column k = last is A069513.
%Y A386577 Row sums are A168324 or A008480.
%Y A386577 The number of trailing zeros in each row is A297155 = A001221-1.
%Y A386577 Column k = 1 is A335452.
%Y A386577 The number of leading zeros in each row is A374246.
%Y A386577 For separability instead of inseparability we have A374252.
%Y A386577 For a multiset with prescribed multiplicities we have A386578, separability A386579.
%Y A386577 A003242 and A335452 count anti-runs, ranks A333489, patterns A005649.
%Y A386577 A025065(n - 2) counts partitions of inseparable type, ranks A335126, sums of A386586.
%Y A386577 A124762 gives inseparability of standard compositions, separability A333382.
%Y A386577 A325534 counts separable multisets, ranks A335433, sums of A386583.
%Y A386577 A325535 counts inseparable multisets, ranks A335448, sums of A386584.
%Y A386577 A336106 counts partitions of separable type, ranks A335127, sums of A386585.
%Y A386577 Cf. A001221, A051903, A051904, A106351, A238130, A336102, A373957, A382525, A386581, A386632.
%K A386577 nonn,tabf
%O A386577 1,6
%A A386577 _Gus Wiseman_, Aug 01 2025