cp's OEIS Frontend

A381454 Number of multisets that can be obtained by choosing a strict integer partition of each prime index of n and taking the multiset union.

%I A381454 #6 Mar 09 2025 12:29:13
%S A381454 1,1,1,1,2,1,2,1,1,2,3,1,4,2,2,1,5,1,6,2,2,3,8,1,3,4,1,2,10,2,12,1,3,
%T A381454 5,4,1,15,6,4,2,18,2,22,3,2,8,27,1,3,3,5,4,32,1,6,2,6,10,38,2,46,12,2,
%U A381454 1,8,3,54,5,8,4,64,1,76,15,3,6,6,4,89,2,1
%N A381454 Number of multisets that can be obtained by choosing a strict integer partition of each prime index of n and taking the multiset union.
%C A381454 First differs from A357982 at a(25) = 3, A357982(25) = 4.
%C A381454 A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
%C A381454 A multiset partition can be regarded as an arrow in the ranked poset of integer partitions. For example, we have {{1},{1,2},{1,3},{1,2,3}}: {1,1,1,1,2,2,3,3} -> {1,3,4,6}, or (33221111) -> (6431) (depending on notation).
%C A381454 Set multipartitions are generally not transitive. For example, we have arrows: {{1},{1,2}}: {1,1,2} -> {1,3} and {{1,3}}: {1,3} -> {4}, but there is no set multipartition {1,1,2} -> {4}.
%F A381454 a(A002110(n)) = A381808(n).
%e A381454 The a(25) = 3 multisets are: {3,3}, {1,2,3}, {1,1,2,2}.
%t A381454 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A381454 Table[Length[Union[Sort/@Join@@@Tuples[Select[IntegerPartitions[#],UnsameQ@@#&]&/@prix[n]]]],{n,100}]
%Y A381454 For constant instead of strict partitions see A381453, A355733, A381455, A000688.
%Y A381454 Positions of 1 are A003586.
%Y A381454 The upper version is A381078, before sums A050320.
%Y A381454 For distinct block-sums see A381634, A381633, A381806.
%Y A381454 Multiset partitions of prime indices:
%Y A381454 - For multiset partitions (A001055) see A317141 (upper), A300383 (lower).
%Y A381454 - For strict multiset partitions (A045778) see A381452.
%Y A381454 - For set systems (A050326, zeros A293243) see A381441 (upper).
%Y A381454 - For sets of constant multisets (A050361) see A381715.
%Y A381454 - For strict multiset partitions with distinct sums (A321469) see A381637.
%Y A381454 - For sets of constant multisets with distinct sums (A381635, zeros A381636) see A381716.
%Y A381454 More on set systems: A050342, A116539, A296120, A318361.
%Y A381454 More on set multipartitions: A089259, A116540, A270995, A296119, A318360.
%Y A381454 More on set multipartitions with distinct sums: A279785, A381717, A381718.
%Y A381454 A000041 counts integer partitions, strict A000009.
%Y A381454 A000040 lists the primes.
%Y A381454 A003963 gives product of prime indices.
%Y A381454 A055396 gives least prime index, greatest A061395.
%Y A381454 A056239 adds up prime indices, row sums of A112798.
%Y A381454 A122111 represents conjugation in terms of Heinz numbers.
%Y A381454 A265947 counts refinement-ordered pairs of integer partitions.
%Y A381454 A358914 counts twice-partitions into distinct strict partitions.
%Y A381454 Cf. A000720, A001222, A002846, A005117, A066328, A213242, A213385, A213427, A293511, A299200, A299201, A299202, A300385, A317142.
%K A381454 nonn
%O A381454 1,5
%A A381454 _Gus Wiseman_, Mar 08 2025