cp's OEIS Frontend

A382427 Number of integer partitions of n that can be partitioned into constant blocks with distinct sums.

%I A382427 #13 Apr 27 2025 09:09:21
%S A382427 1,1,2,3,4,7,11,14,19,28,39,50,70,91,120,161,203,260,338,426,556,695,
%T A382427 863,1082,1360,1685
%N A382427 Number of integer partitions of n that can be partitioned into constant blocks with distinct sums.
%C A382427 Conjecture: Also the number of integer partitions of n having a permutation with all distinct run-sums.
%e A382427 The partition (3,2,2,2,1) can be partitioned as {{1},{2},{3},{2,2}} or {{1},{3},{2,2,2}}, so is counted under a(10).
%e A382427 The a(1) = 1 through a(7) = 14 partitions:
%e A382427   (1)  (2)   (3)    (4)     (5)      (6)       (7)
%e A382427        (11)  (21)   (22)    (32)     (33)      (43)
%e A382427              (111)  (31)    (41)     (42)      (52)
%e A382427                     (1111)  (221)    (51)      (61)
%e A382427                             (311)    (222)     (322)
%e A382427                             (2111)   (321)     (331)
%e A382427                             (11111)  (411)     (421)
%e A382427                                      (2211)    (511)
%e A382427                                      (3111)    (2221)
%e A382427                                      (21111)   (4111)
%e A382427                                      (111111)  (22111)
%e A382427                                                (31111)
%e A382427                                                (211111)
%e A382427                                                (1111111)
%t A382427 hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
%t A382427 pfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#,d]&)/@Select[pfacs[n/d],Min@@#>=d&],{d,Select[Rest[Divisors[n]],PrimePowerQ]}]];
%t A382427 Table[Length[Select[IntegerPartitions[n],Select[pfacs[Times@@Prime/@#],UnsameQ@@hwt/@#&]!={}&]],{n,0,10}]
%Y A382427 Twice-partitions of this type (constant with distinct) are counted by A279786.
%Y A382427 Multiset partitions of this type are ranked by A326535 /\ A355743.
%Y A382427 The complement is counted by A381717, ranks A381636, zeros of A381635.
%Y A382427 For strict instead of constant blocks we have A381992, ranks A382075.
%Y A382427 For a unique choice we have A382301, ranks A381991.
%Y A382427 Normal multiset partitions of this type are counted by A382203, sets A381718.
%Y A382427 A000041 counts integer partitions, strict A000009.
%Y A382427 A000688 counts factorizations into prime powers, see A381455, A381453.
%Y A382427 A001055 counts factorizations, strict A045778, see A317141, A300383, A265947.
%Y A382427 A050361 counts factorizations into distinct prime powers.
%Y A382427 Cf. A006171, A047966, A279784, A295935, A300385, A353864, A381633, A381716, A381990, A381993, A382079, A382876.
%K A382427 nonn,more
%O A382427 0,3
%A A382427 _Gus Wiseman_, Mar 26 2025