A371788 - OEIS

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 #7 Apr 16 2024 23:37:49

%S 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 363,297,93,12,1,0,12,248,1292,1660,753,158,16,1,0,11,563,4649,8409,

%U 5591,1653,250,20,1,0,2,1414,15976,41264,38074,15590,3249,380,25,1

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

%e The set partition {{1,3},{2},{4}} has two distinct block-sums {2,4} so is counted under T(4,2).

%e Triangle begins:

%e 1

%e 0 1

%e 0 1 1

%e 0 2 2 1

%e 0 2 8 4 1

%e 0 2 19 24 6 1

%e 0 2 47 95 49 9 1

%e 0 6 105 363 297 93 12 1

%e 0 12 248 1292 1660 753 158 16 1

%e 0 11 563 4649 8409 5591 1653 250 20 1

%e 0 2 1414 15976 41264 38074 15590 3249 380 25 1

%e Row n = 4 counts the following set partitions:

%e . {{1,4},{2,3}} {{1},{2,3,4}} {{1},{2},{3,4}} {{1},{2},{3},{4}}

%e {{1,2,3,4}} {{1,2},{3},{4}} {{1},{2,3},{4}}

%e {{1,2},{3,4}} {{1},{2,4},{3}}

%e {{1,3},{2},{4}} {{1,4},{2},{3}}

%e {{1,3},{2,4}}

%e {{1,2,3},{4}}

%e {{1,2,4},{3}}

%e {{1,3,4},{2}}

%t sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];

%t Table[Length[Select[sps[Range[n]], Length[Union[Total/@#]]==k&]],{n,0,5},{k,0,n}]

%Y Row sums are A000110.

%Y Column k = 1 is A035470.

%Y A version for integer partitions is A116608.

%Y For block lengths instead of sums we have A208437.

%Y A008277 counts set partitions by length.

%Y A275780 counts set partitions with distinct block-sums.

%Y A371737 counts quanimous strict partitions, non-strict A321452.

%Y A371789 counts non-quanimous sets, differences A371790.

%Y Cf. A007837, A038041, A327899, A336137, A336138, A365663, A365661, A365925.

%K nonn,tabl

%O 0,8

%A _Gus Wiseman_, Apr 16 2024