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A371788 Triangle read by rows where T(n,k) is the number of set partitions of {1..n} with exactly k distinct block-sums.

%I A371788 #7 Apr 16 2024 23:37:49
%S A371788 1,0,1,0,1,1,0,2,2,1,0,2,8,4,1,0,2,19,24,6,1,0,2,47,95,49,9,1,0,6,105,
%T A371788 363,297,93,12,1,0,12,248,1292,1660,753,158,16,1,0,11,563,4649,8409,
%U A371788 5591,1653,250,20,1,0,2,1414,15976,41264,38074,15590,3249,380,25,1
%N A371788 Triangle read by rows where T(n,k) is the number of set partitions of {1..n} with exactly k distinct block-sums.
%e A371788 The set partition {{1,3},{2},{4}} has two distinct block-sums {2,4} so is counted under T(4,2).
%e A371788 Triangle begins:
%e A371788      1
%e A371788      0     1
%e A371788      0     1     1
%e A371788      0     2     2     1
%e A371788      0     2     8     4     1
%e A371788      0     2    19    24     6     1
%e A371788      0     2    47    95    49     9     1
%e A371788      0     6   105   363   297    93    12     1
%e A371788      0    12   248  1292  1660   753   158    16     1
%e A371788      0    11   563  4649  8409  5591  1653   250    20     1
%e A371788      0     2  1414 15976 41264 38074 15590  3249   380    25     1
%e A371788 Row n = 4 counts the following set partitions:
%e A371788   .  {{1,4},{2,3}}  {{1},{2,3,4}}    {{1},{2},{3,4}}  {{1},{2},{3},{4}}
%e A371788      {{1,2,3,4}}    {{1,2},{3},{4}}  {{1},{2,3},{4}}
%e A371788                     {{1,2},{3,4}}    {{1},{2,4},{3}}
%e A371788                     {{1,3},{2},{4}}  {{1,4},{2},{3}}
%e A371788                     {{1,3},{2,4}}
%e A371788                     {{1,2,3},{4}}
%e A371788                     {{1,2,4},{3}}
%e A371788                     {{1,3,4},{2}}
%t A371788 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A371788 Table[Length[Select[sps[Range[n]], Length[Union[Total/@#]]==k&]],{n,0,5},{k,0,n}]
%Y A371788 Row sums are A000110.
%Y A371788 Column k = 1 is A035470.
%Y A371788 A version for integer partitions is A116608.
%Y A371788 For block lengths instead of sums we have A208437.
%Y A371788 A008277 counts set partitions by length.
%Y A371788 A275780 counts set partitions with distinct block-sums.
%Y A371788 A371737 counts quanimous strict partitions, non-strict A321452.
%Y A371788 A371789 counts non-quanimous sets, differences A371790.
%Y A371788 Cf. A007837, A038041, A327899, A336137, A336138, A365663, A365661, A365925.
%K A371788 nonn,tabl
%O A371788 0,8
%A A371788 _Gus Wiseman_, Apr 16 2024