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A386636 Triangle read by rows where T(n,k) is the number of inseparable type set partitions of {1..n} into k blocks.

%I A386636 #6 Aug 10 2025 22:00:25
%S A386636 0,0,0,0,1,0,0,1,0,0,0,1,4,0,0,0,1,5,0,0,0,0,1,21,15,0,0,0,0,1,28,21,
%T A386636 0,0,0,0,0,1,92,196,56,0,0,0,0,0,1,129,288,84,0,0,0,0,0,0,1,385,1875,
%U A386636 1380,210,0,0,0,0,0,0,1,561,2860,2145,330,0,0,0,0,0,0
%N A386636 Triangle read by rows where T(n,k) is the number of inseparable type set partitions of {1..n} into k blocks.
%C A386636 A set partition is of inseparable type iff the underlying set has no permutation whose adjacent elements always belong to different blocks. Note that this only depends on the sizes of the blocks.
%C A386636 A set partition is also of inseparable type iff its greatest block size is at least 2 more than the sum of all its other block sizes.
%C A386636 This is different from inseparable partitions (A325535) and partitions of inseparable type (A386638 or A025065).
%e A386636 Row n = 6 counts the following set partitions:
%e A386636   .  {123456}  {1}{23456}  {1}{2}{3456}  .  .  .
%e A386636                {12}{3456}  {1}{2345}{6}
%e A386636                {13}{2456}  {1}{2346}{5}
%e A386636                {14}{2356}  {1}{2356}{4}
%e A386636                {15}{2346}  {1}{2456}{3}
%e A386636                {16}{2345}  {1234}{5}{6}
%e A386636                {1234}{56}  {1235}{4}{6}
%e A386636                {1235}{46}  {1236}{4}{5}
%e A386636                {1236}{45}  {1245}{3}{6}
%e A386636                {1245}{36}  {1246}{3}{5}
%e A386636                {1246}{35}  {1256}{3}{4}
%e A386636                {1256}{34}  {1345}{2}{6}
%e A386636                {1345}{26}  {1346}{2}{5}
%e A386636                {1346}{25}  {1356}{2}{4}
%e A386636                {1356}{24}  {1456}{2}{3}
%e A386636                {1456}{23}
%e A386636                {12345}{6}
%e A386636                {12346}{5}
%e A386636                {12356}{4}
%e A386636                {12456}{3}
%e A386636                {13456}{2}
%e A386636 Triangle begins:
%e A386636     0
%e A386636     0    0
%e A386636     0    1    0
%e A386636     0    1    0    0
%e A386636     0    1    4    0    0
%e A386636     0    1    5    0    0    0
%e A386636     0    1   21   15    0    0    0
%e A386636     0    1   28   21    0    0    0    0
%e A386636     0    1   92  196   56    0    0    0    0
%e A386636     0    1  129  288   84    0    0    0    0    0
%e A386636     0    1  385 1875 1380  210    0    0    0    0    0
%t A386636 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A386636 stnseps[stn_]:=Select[Permutations[Union@@stn],And@@Table[Position[stn,#[[i]]][[1,1]]!=Position[stn,#[[i+1]]][[1,1]],{i,Length[#]-1}]&]
%t A386636 Table[Length[Select[sps[Range[n]],Length[#]==k&&stnseps[#]=={}&]],{n,0,5},{k,0,n}]
%Y A386636 For separable partitions we have A386583, sums A325534, ranks A335433.
%Y A386636 For inseparable partitions we have A386584, sums A325535, ranks A335448.
%Y A386636 For separable type partitions we have A386585, sums A336106, ranks A335127.
%Y A386636 For inseparable type partitions we have A386586, sums A386638 or A025065, ranks A335126.
%Y A386636 Row sums are A386634.
%Y A386636 The complement is counted by A386635, row sums A386633.
%Y A386636 A000110 counts set partitions, row sums of A048993.
%Y A386636 A000670 counts ordered set partitions.
%Y A386636 A003242 and A335452 count anti-runs, ranks A333489, patterns A005649.
%Y A386636 A279790 counts disjoint families on strongly normal multisets.
%Y A386636 A335434 counts separable factorizations, inseparable A333487.
%Y A386636 A336103 counts normal separable multisets, inseparable A336102.
%Y A386636 A386587 counts disjoint families of strict partitions of each prime exponent.
%Y A386636 Cf. A008284, A072233, A097805, A106351, A116861, A124762, A129506, A190945, A239455, A335125, A374252, A386575.
%K A386636 nonn,tabl
%O A386636 0,13
%A A386636 _Gus Wiseman_, Aug 10 2025