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A386635 Triangle read by rows where T(n,k) is the number of separable type set partitions of {1..n} into k blocks.

%I A386635 #6 Aug 10 2025 22:00:20
%S A386635 1,0,1,0,0,1,0,0,3,1,0,0,3,6,1,0,0,10,25,10,1,0,0,10,75,65,15,1,0,0,
%T A386635 35,280,350,140,21,1,0,0,35,770,1645,1050,266,28,1,0,0,126,2737,7686,
%U A386635 6951,2646,462,36,1,0,0,126,7455,32725,42315,22827,5880,750,45,1
%N A386635 Triangle read by rows where T(n,k) is the number of separable type set partitions of {1..n} into k blocks.
%C A386635 A set partition is of separable type iff the underlying set has a permutation whose adjacent elements always belong to different blocks. Note that this only depends on the sizes of the blocks.
%C A386635 A set partition is also of separable type iff its greatest block size is at most one more than the sum of all its other blocks sizes.
%C A386635 This is different from separable partitions (A325534) and partitions of separable type (A336106).
%e A386635 Row n = 4 counts the following set partitions:
%e A386635   .  .  {{1,2},{3,4}}  {{1},{2},{3,4}}  {{1},{2},{3},{4}}
%e A386635         {{1,3},{2,4}}  {{1},{2,3},{4}}
%e A386635         {{1,4},{2,3}}  {{1},{2,4},{3}}
%e A386635                        {{1,2},{3},{4}}
%e A386635                        {{1,3},{2},{4}}
%e A386635                        {{1,4},{2},{3}}
%e A386635 Triangle begins:
%e A386635     1
%e A386635     0    1
%e A386635     0    0    1
%e A386635     0    0    3    1
%e A386635     0    0    3    6    1
%e A386635     0    0   10   25   10    1
%e A386635     0    0   10   75   65   15    1
%e A386635     0    0   35  280  350  140   21    1
%t A386635 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A386635 stnseps[stn_]:=Select[Permutations[Union@@stn],And@@Table[Position[stn,#[[i]]][[1,1]]!=Position[stn,#[[i+1]]][[1,1]],{i,Length[#]-1}]&];
%t A386635 Table[Length[Select[sps[Range[n]],Length[#]==k&&stnseps[#]!={}&]],{n,0,5},{k,0,n}]
%Y A386635 Column k = 2 appears to be A128015.
%Y A386635 For separable partitions we have A386583, sums A325534, ranks A335433.
%Y A386635 For inseparable partitions we have A386584, sums A325535, ranks A335448.
%Y A386635 For separable type partitions we have A386585, sums A336106, ranks A335127.
%Y A386635 For inseparable type partitions we have A386586, sums A386638 or A025065, ranks A335126.
%Y A386635 Row sums are A386633.
%Y A386635 The complement is counted by A386636, row sums A386634.
%Y A386635 A000110 counts set partitions, row sums of A048993.
%Y A386635 A000670 counts ordered set partitions.
%Y A386635 A003242 and A335452 count anti-runs, ranks A333489, patterns A005649.
%Y A386635 A239455 counts Look-and-Say partitions, ranks A351294, conjugate A381432.
%Y A386635 A335434 counts separable factorizations, inseparable A333487.
%Y A386635 A336103 counts normal separable multisets, inseparable A336102.
%Y A386635 A351293 counts non-Look-and-Say partitions, ranks A351295, conjugate A381433.
%Y A386635 A386587 counts disjoint families of strict partitions of each prime exponent.
%Y A386635 Cf. A072233, A097805, A106351, A116861, A190945, A279790, A333382, A335125, A374252, A386575, A386578.
%K A386635 nonn,tabl
%O A386635 0,9
%A A386635 _Gus Wiseman_, Aug 10 2025