cp's OEIS Frontend

A386634 Number of inseparable type set partitions of {1..n}.

%I A386634 #14 Aug 11 2025 15:09:21
%S A386634 0,0,1,1,5,6,37,50,345,502,3851,5897,49854,79249,730745,1195147,
%T A386634 11915997,19929390,213332101,363275555,4150104224,7172334477,
%U A386634 87003759195,152231458128,1952292972199,3451893361661,46625594567852,83183249675125,1179506183956655,2120758970878892
%N A386634 Number of inseparable type set partitions of {1..n}.
%C A386634 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 A386634 A set partition is also of inseparable type iff its greatest block size is at least 2 more than the sum of its other block sizes.
%C A386634 This is different from inseparable partitions (A325535) and partitions of inseparable type (A386638 or A025065).
%H A386634 Alois P. Heinz, <a href="/A386634/b386634.txt">Table of n, a(n) for n = 0..250</a>
%e A386634 The a(2) = 1 through a(5) = 6 set partitions:
%e A386634   {{1,2}}  {{1,2,3}}  {{1,2,3,4}}    {{1,2,3,4,5}}
%e A386634                       {{1},{2,3,4}}  {{1},{2,3,4,5}}
%e A386634                       {{1,2,3},{4}}  {{1,2,3,4},{5}}
%e A386634                       {{1,2,4},{3}}  {{1,2,3,5},{4}}
%e A386634                       {{1,3,4},{2}}  {{1,2,4,5},{3}}
%e A386634                                      {{1,3,4,5},{2}}
%t A386634 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A386634 stnseps[stn_]:=Select[Permutations[Union@@stn],And@@Table[Position[stn,#[[i]]][[1,1]]!=Position[stn,#[[i+1]]][[1,1]],{i,Length[#]-1}]&]
%t A386634 Table[Length[Select[sps[Range[n]],stnseps[#]=={}&]],{n,0,5}]
%Y A386634 For separable partitions we have A386583, sums A325534, ranks A335433.
%Y A386634 For inseparable partitions we have A386584, sums A325535, ranks A335448.
%Y A386634 For separable type partitions we have A386585, sums A336106, ranks A335127.
%Y A386634 For inseparable type partitions we have A386586, sums A386638 or A025065, ranks A335126.
%Y A386634 The complement is counted by A386633, sums of A386635.
%Y A386634 Row sums of A386636.
%Y A386634 A000110 counts set partitions, row sums of A048993.
%Y A386634 A000670 counts ordered set partitions.
%Y A386634 A003242 and A335452 count anti-runs, ranks A333489, patterns A005649.
%Y A386634 A279790 counts disjoint families on strongly normal multisets.
%Y A386634 A335434 counts separable factorizations, inseparable A333487.
%Y A386634 A336103 counts normal separable multisets, inseparable A336102.
%Y A386634 A386587 counts disjoint families of strict partitions of each prime exponent.
%Y A386634 Cf. A001055, A124762, A239455, A335125, A386575, A386579.
%K A386634 nonn
%O A386634 0,5
%A A386634 _Gus Wiseman_, Aug 09 2025
%E A386634 a(12)-a(29) from _Alois P. Heinz_, Aug 10 2025