A383519 - OEIS

A383519

Number of section-sum partitions of n that have all distinct multiplicities (Wilf).

1, 1, 2, 2, 3, 3, 6, 7, 9, 12, 14, 19, 21, 27, 30, 33, 41, 50, 57, 68, 79, 89, 112, 126, 144, 172, 198, 220, 257, 298, 327, 383, 423, 477, 533, 621, 650, 760, 816, 920, 1013

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OFFSET

0,3

COMMENTS

An integer partition is section-sum iff it is possible to choose a disjoint family of strict partitions, one of each of its positive 0-appended differences. These are ranked by A381432.

An integer partition is Wilf iff its multiplicities are all different (ranked by A130091).

LINKS

Table of n, a(n) for n=0..40.

EXAMPLE

The a(1) = 1 through a(8) = 9 partitions:

(1) (2) (3) (4) (5) (6) (7) (8)

(11) (111) (22) (311) (33) (322) (44)

(1111) (11111) (222) (331) (332)

(411) (511) (611)

(3111) (4111) (2222)

(111111) (31111) (5111)

(1111111) (41111)

(311111)

(11111111)

MATHEMATICA

disjointFamilies[y_]:=Select[Tuples[IntegerPartitions/@Length/@Split[y]], UnsameQ@@Join@@#&];

prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];

conj[y_]:=If[Length[y]==0, y, Table[Length[Select[y, #>=k&]], {k, 1, Max[y]}]];

Table[Length[Select[IntegerPartitions[n], disjointFamilies[conj[#]]!={}&&UnsameQ@@Length/@Split[#]&]], {n, 0, 15}]

CROSSREFS

Ranking sequences are shown in parentheses below.

For Look-and-Say instead of section-sum we have A098859 (A130091), conjugate (A383512).

For non Wilf instead of Wilf we have A383506 (A383514).

These partitions are ranked by (A383520).

A000041 counts integer partitions, strict A000009.

A098859 counts Wilf partitions (A130091), conjugate (A383512).

A239455 counts Look-and-Say partitions (A351294), complement A351293 (A351295).

A239455 counts section-sum partitions (A381432), complement A351293 (A381433).

A336866 counts non Wilf partitions (A130092), conjugate (A383513).

Cf. A047966, A320348, A325324, A325325, A351592, A353837, A381431, A383530, A383709, A384006.

Sequence in context: A054172 A236971 A383507 * A383508 A373446 A121211

Adjacent sequences: A383516 A383517 A383518 * A383520 A383521 A383522

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, May 19 2025

STATUS

approved