cp's OEIS Frontend

A386586 Triangle read by rows where T(n,k) is the number of integer partitions y of n into k parts such that any multiset whose multiplicities are the parts of y is inseparable.

%I A386586 #9 Aug 09 2025 10:11:00
%S A386586 0,0,0,0,1,0,0,1,0,0,0,1,1,0,0,0,1,1,0,0,0,0,1,2,1,0,0,0,0,1,2,1,0,0,
%T A386586 0,0,0,1,3,2,1,0,0,0,0,0,1,3,2,1,0,0,0,0,0,0,1,4,4,2,1,0,0,0,0,0
%N A386586 Triangle read by rows where T(n,k) is the number of integer partitions y of n into k parts such that any multiset whose multiplicities are the parts of y is inseparable.
%C A386586 We say that such partitions are of inseparable type. This is different from inseparable partitions (see A386584). A multiset is separable iff it has a permutation without any adjacent equal parts.
%F A386586 a(n) = A072233(n) - A386585(n).
%e A386586 The partition y = (7,2,1) is the multiplicities of the multiset {1,1,1,1,1,1,1,2,2,3}, which is inseparable, so y is counted under T(10,3).
%e A386586 Row n = 10 counts the following partitions (A = 10):
%e A386586   .  A  91  811  7111  61111  .  .  .  .  .
%e A386586         82  721  6211
%e A386586         73  631
%e A386586         64  622
%e A386586 Triangle begins:
%e A386586   0
%e A386586   0 0
%e A386586   0 1 0
%e A386586   0 1 0 0
%e A386586   0 1 1 0 0
%e A386586   0 1 1 0 0 0
%e A386586   0 1 2 1 0 0 0
%e A386586   0 1 2 1 0 0 0 0
%e A386586   0 1 3 2 1 0 0 0 0
%e A386586   0 1 3 2 1 0 0 0 0 0
%e A386586   0 1 4 4 2 1 0 0 0 0 0
%t A386586 insepQ[y_]:=Select[Permutations[y],Length[Split[#]]==Length[y]&]=={};
%t A386586 ptm[y_]:=Join@@Table[ConstantArray[k,y[[k]]],{k,Length[y]}];
%t A386586 Table[Length[Select[IntegerPartitions[n,{k}],insepQ[ptm[#]]&]],{n,0,5},{k,0,n}]
%Y A386586 This is the inseparable type case of A008284 or A072233.
%Y A386586 Row sums shifted left once are A025065 (ranks A335126), separable version A336106 (ranks A335127).
%Y A386586 For separable instead of inseparable type we have A386583.
%Y A386586 For integer partitions instead of normal multisets we have A386584.
%Y A386586 For separable type instead of inseparable type we have A386585.
%Y A386586 A003242 and A335452 count anti-runs, ranks A333489, patterns A005649.
%Y A386586 A239455 counts Look-and-Say partitions, ranks A351294.
%Y A386586 A325534 counts separable multisets, ranks A335433.
%Y A386586 A325535 counts inseparable multisets, ranks A335448.
%Y A386586 A336103 counts normal separable multisets, inseparable A336102.
%Y A386586 A351293 counts non-Look-and-Say partitions, ranks A351295.
%Y A386586 Cf. A000670, A005651, A106351, A238130, A279790, A386575, A386576, A386579, A386582, A386634.
%K A386586 nonn,tabl,more
%O A386586 0,24
%A A386586 _Gus Wiseman_, Aug 05 2025