cp's OEIS Frontend

A381717 Number of integer partitions of n that cannot be partitioned into constant multisets with distinct block-sums.

%I A381717 #15 Apr 27 2025 09:09:10
%S A381717 0,0,0,0,1,0,0,1,3,2,3,6,7,10,15,15,28,37,47,64,71,97,139,173,215,273,
%T A381717 361,439,551,691,853,1078,1325,1623,2046,2458,2998,3697,4527,5472,
%U A381717 6590,7988,9590,11598,13933,16560,19976,23822,28420,33797,40088,47476,56369,66678
%N A381717 Number of integer partitions of n that cannot be partitioned into constant multisets with distinct block-sums.
%C A381717 Conjecture: Also the number of integer partitions of n having no permutation with all distinct run-sums, ranked by zeros of A382876. In other words, a partition has a permutation with all distinct run-sums iff it has a multiset partition into constant blocks with all distinct block-sums, where the run-sums of a sequence are obtained by splitting it into maximal runs and taking their sums.
%e A381717 For y = (3,2,2,1) we have the multiset partition {{3},{2,2},{1}}, so y is not counted under a(8).
%e A381717 For y = (3,2,1,1,1) there are 3 multiset partitions into constant multisets:
%e A381717   {{3},{2},{1,1,1}}
%e A381717   {{3},{2},{1,1},{1}}
%e A381717   {{3},{2},{1},{1},{1}}
%e A381717 but none of these has distinct block-sums, so y is counted under a(8).
%e A381717 For y = (3,3,1,1,1,1,1,1) we have multiset partitions:
%e A381717   {{1},{3,3},{1,1,1,1,1}}
%e A381717   {{1,1},{3,3},{1,1,1,1}}
%e A381717   {{1},{1,1},{3,3},{1,1,1}}
%e A381717 so y is not counted under a(12).
%e A381717 The a(4) = 1 through a(13) = 10 partitions:
%e A381717   211  .  .  3211  422    4221  6211   4322     633      5422
%e A381717                    4211   5211  33211  7211     8211     6331
%e A381717                    32111        42211  43211    43221    9211
%e A381717                                        422111   44211    54211
%e A381717                                        431111   53211    63211
%e A381717                                        3221111  432111   333211
%e A381717                                                 4221111  432211
%e A381717                                                          532111
%e A381717                                                          4321111
%e A381717                                                          42211111
%t A381717 mce[y_]:=Table[ConstantArray[y[[1]],#]&/@ptn,{ptn,IntegerPartitions[Length[y]]}];
%t A381717 Table[Length[Select[IntegerPartitions[n],Select[Join@@@Tuples[mce/@Split[#]],UnsameQ@@Total/@#&]=={}&]],{n,0,30}]
%Y A381717 Twice-partitions of this type (constant with distinct) are counted by A279786.
%Y A381717 Multiset partitions of this type are ranked by A326535 /\ A355743.
%Y A381717 These partitions are ranked by A381636, zeros of A381635.
%Y A381717 For strict instead of constant blocks we have A381990, see A381806, A381633, A382079.
%Y A381717 For equal instead of distinct block-sums we have A381993.
%Y A381717 A000041 counts integer partitions, strict A000009.
%Y A381717 A000688 counts factorizations into prime powers, see A381455, A381453.
%Y A381717 A001055 counts factorizations, strict A045778, see A317141, A300383.
%Y A381717 A050361 counts factorizations into distinct prime powers.
%Y A381717 Cf. A002846, A047966, A130091, A213242, A213427, A265947, A317142, A353864, A381716, A381991, A381992, A382876.
%K A381717 nonn
%O A381717 0,9
%A A381717 _Gus Wiseman_, Mar 16 2025
%E A381717 a(37)-a(53) from _Robert Price_, Mar 31 2025