A383508 - OEIS

A383508

Number of integer partitions of n that are both Look-and-Say and section-sum partitions.

1, 1, 2, 2, 3, 3, 6, 7, 9, 12, 14, 19, 22, 27, 30, 35, 42, 50, 58, 68, 82, 92, 112, 126, 149, 174, 199, 225, 263, 299, 337, 388, 435, 488, 545, 635, 681, 775, 841, 948, 1051, 1181, 1271, 1446, 1553, 1765, 1896, 2141, 2285, 2608, 2799

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OFFSET

0,3

COMMENTS

An integer partition is Look-and-Say iff it is possible to choose a disjoint family of strict partitions, one of each of its multiplicities. These are ranked by A351294.

An integer partition is section-sum iff its conjugate is Look-and-Say, meaning 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.

LINKS

Table of n, a(n) for n=0..50.

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@@#&];

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

Table[Length[Select[IntegerPartitions[n], disjointFamilies[#]!={}&&disjointFamilies[conj[#]]!={}&]], {n, 0, 30}]

CROSSREFS

Ranking sequences are shown in parentheses below.

The non Wilf case is A383511 (A383518).

These partitions are ranked by (A383515).

A000041 counts integer partitions, strict A000009.

A047993 counts partitions with max part = length (A106529).

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

A122111 represents conjugation in terms of Heinz numbers.

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).

A351592 counts non Wilf Look-and-Say partitions (A384006).

A383509 counts partitions that are Look-and-Say but not section-sum (A383516).

A383509 counts partitions that are not Look-and-Say but are section-sum (A384007).

A383510 counts partitions that are neither Look-and-Say nor section-sum (A383517).

Cf. A033461, A047966, A048767, A320348, A325324, A325325, A381431, A383519, A383530, A383709.

Sequence in context: A236971 A383507 A383519 * A373446 A121211 A316585

Adjacent sequences: A383505 A383506 A383507 * A383509 A383510 A383511

KEYWORD

nonn

AUTHOR

Gus Wiseman, May 17 2025

STATUS

approved