cp's OEIS Frontend

A386585 Triangle read by rows where T(n,k) is the number of integer partitions y of n into k = 0..n parts such that any multiset whose multiplicities are the parts of y is separable.

%I A386585 #16 Aug 09 2025 10:10:54
%S A386585 1,0,1,0,0,1,0,0,1,1,0,0,1,1,1,0,0,1,2,1,1,0,0,1,2,2,1,1,0,0,1,3,3,2,
%T A386585 1,1,0,0,1,3,4,3,2,1,1,0,0,1,5,5,5,3,2,1,1,0,0,1,4,7,6,5,3,2,1,1
%N A386585 Triangle read by rows where T(n,k) is the number of integer partitions y of n into k = 0..n parts such that any multiset whose multiplicities are the parts of y is separable.
%C A386585 We say that such partitions are of separable type.
%C A386585 A multiset is separable iff it has a permutation without any adjacent equal parts.
%F A386585 a(n) = A072233(n) - A386586(n).
%e A386585 Row n = 8 counts the following partitions:
%e A386585   .  .  44  431  4211  41111  311111  2111111  11111111
%e A386585             422  3311  32111  221111
%e A386585             332  3221  22211
%e A386585                  2222
%e A386585 with the following separable multisets:
%e A386585   . . 11112222 11112223 11112234 11112345 11123456 11234567 12345678
%e A386585                11112233 11122234 11122345 11223456
%e A386585                11122233 11122334 11223345
%e A386585                         11223344
%e A386585 Triangle begins:
%e A386585   1
%e A386585   0  1
%e A386585   0  0  1
%e A386585   0  0  1  1
%e A386585   0  0  1  1  1
%e A386585   0  0  1  2  1  1
%e A386585   0  0  1  2  2  1  1
%e A386585   0  0  1  3  3  2  1  1
%e A386585   0  0  1  3  4  3  2  1  1
%e A386585   0  0  1  5  5  5  3  2  1  1
%e A386585   0  0  1  4  7  6  5  3  2  1  1
%t A386585 sepQ[y_]:=Select[Permutations[y],Length[Split[#]]==Length[y]&]!={};
%t A386585 mst[y_]:=Join@@Table[ConstantArray[k,y[[k]]],{k,Length[y]}];
%t A386585 Table[Length[Select[IntegerPartitions[n,{k}],sepQ[mst[#]]&]],{n,0,5},{k,0,n}]
%Y A386585 This is the separable type case of A072233 or A008284.
%Y A386585 Row sums are A336106, ranks A335127.
%Y A386585 For separable instead of separable type we have A386583, inseparable A386584.
%Y A386585 For inseparable instead of separable we have A386586, sums A025065, ranks A335126.
%Y A386585 A003242 and A335452 count anti-runs, ranks A333489, patterns A005649.
%Y A386585 A239455 counts Look-and-Say partitions, ranks A351294.
%Y A386585 A279790 counts disjoint families on strongly normal multisets.
%Y A386585 A325534 counts separable multisets, ranks A335433.
%Y A386585 A325535 counts inseparable multisets, ranks A335448.
%Y A386585 A336103 counts normal separable multisets, inseparable A336102.
%Y A386585 A351293 counts non-Look-and-Say partitions, ranks A351295.
%Y A386585 A386633 counts separable set partitions, row sums of A386635.
%Y A386585 A386634 counts inseparable set partitions, row sums of A386636.
%Y A386585 Cf. A005651, A106351, A111133, A238130, A335434, A386575, A386576, A386579, A386580, A386581, A386582.
%K A386585 nonn,tabl,more
%O A386585 0,19
%A A386585 _Gus Wiseman_, Aug 02 2025