cp's OEIS Frontend

A371789 Number of non-quanimous subsets of {1..n}, meaning there is only one set partition with all equal block-sums.

%I A371789 #14 Mar 30 2025 04:28:24
%S A371789 1,2,4,7,13,24,45,85,162,306,585,1102,2106,3988,7623,14535,27758,
%T A371789 52921,101848,195618,378383,733609,1421868,2755807,5373060,10482925,
%U A371789 20495335,40119622,78476107,153463714,300732073
%N A371789 Number of non-quanimous subsets of {1..n}, meaning there is only one set partition with all equal block-sums.
%C A371789 A finite multiset of numbers is defined to be quanimous iff it can be partitioned into two or more multisets with equal sums. Quanimous partitions are counted by A321452 and ranked by A321454.
%e A371789 The set s = {3,4,6,8,9} has set partitions {{3,4,6,8,9}} and {{3,4,8},{6,9}} with equal block-sums, so s is not counted under a(9).
%e A371789 The a(0) = 1 through a(4) = 13 subsets:
%e A371789   {}  {}   {}     {}     {}
%e A371789       {1}  {1}    {1}    {1}
%e A371789            {2}    {2}    {2}
%e A371789            {1,2}  {3}    {3}
%e A371789                   {1,2}  {4}
%e A371789                   {1,3}  {1,2}
%e A371789                   {2,3}  {1,3}
%e A371789                          {1,4}
%e A371789                          {2,3}
%e A371789                          {2,4}
%e A371789                          {3,4}
%e A371789                          {1,2,4}
%e A371789                          {2,3,4}
%t A371789 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A371789 Table[Length[Select[Subsets[Range[n]], Length[Select[sps[#],SameQ@@Total/@#&]]==1&]],{n,0,8}]
%Y A371789 The "bi-" complement for integer partitions is A002219, ranks A357976.
%Y A371789 The "bi-" complement for strict partitions is A237258, ranks A357854.
%Y A371789 The version for integer partitions is A321451, ranks A321453.
%Y A371789 The complement for integer partitions is A321452, ranks A321454
%Y A371789 The version for strict partitions is A371736, complement A371737.
%Y A371789 First differences are A371790.
%Y A371789 The "bi-" version is A371792, complement A371791.
%Y A371789 The "bi-" version for strict partitions is A371794 (bisection A321142).
%Y A371789 The "bi-" version for integer partitions is A371795, ranks A371731.
%Y A371789 The complement is counted by A371796, differences A371797.
%Y A371789 A108917 counts knapsack partitions, ranks A299702, strict A275972.
%Y A371789 A366754 counts non-knapsack partitions, ranks A299729, strict A316402.
%Y A371789 A371783 counts k-quanimous partitions.
%Y A371789 Cf. A000005, A018818, A035470, A038041, A232466, A365663, A365661.
%K A371789 nonn,more
%O A371789 0,2
%A A371789 _Gus Wiseman_, Apr 17 2024
%E A371789 a(11)-a(30) from _Bert Dobbelaere_, Mar 30 2025