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

1, 1, 2, 3, 4, 7, 11, 14, 19, 28, 39, 50, 70, 91, 120, 161, 203, 260, 338, 426, 556, 695, 863, 1082, 1360, 1685

Offset: 0

Author

Gus Wiseman, Mar 26 2025

Keywords

• nonn
• more

Comments

Conjecture: Also the number of integer partitions of n having a permutation with all distinct run-sums.

Examples

			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).
The a(1) = 1 through a(7) = 14 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (21)   (22)    (32)     (33)      (43)
             (111)  (31)    (41)     (42)      (52)
                    (1111)  (221)    (51)      (61)
                            (311)    (222)     (322)
                            (2111)   (321)     (331)
                            (11111)  (411)     (421)
                                     (2211)    (511)
                                     (3111)    (2221)
                                     (21111)   (4111)
                                     (111111)  (22111)
                                               (31111)
                                               (211111)
                                               (1111111)
		

Crossrefs

Twice-partitions of this type (constant with distinct) are counted by A279786.

Multiset partitions of this type are ranked by A326535 /\ A355743.

The complement is counted by A381717, ranks A381636, zeros of A381635.

For strict instead of constant blocks we have A381992, ranks A382075.

For a unique choice we have A382301, ranks A381991.

Normal multiset partitions of this type are counted by A382203, sets A381718.

A000041 counts integer partitions, strict A000009.

A000688 counts factorizations into prime powers, see A381455, A381453.

A001055 counts factorizations, strict A045778, see A317141, A300383, A265947.

A050361 counts factorizations into distinct prime powers.

Cf. A006171, A047966, A279784, A295935, A300385, A353864, A381633, A381716, A381990, A381993, A382079, A382876.

Programs

Mathematica

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
pfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#,d]&)/@Select[pfacs[n/d],Min@@#>=d&],{d,Select[Rest[Divisors[n]],PrimePowerQ]}]];
Table[Length[Select[IntegerPartitions[n],Select[pfacs[Times@@Prime/@#],UnsameQ@@hwt/@#&]!={}&]],{n,0,10}]