A383509 - OEIS

A383509

Number of Look-and-Say partitions of n that are not section-sum partitions.

0, 0, 0, 0, 1, 2, 1, 3, 4, 4, 7, 9, 11, 18, 25, 30, 41, 55, 63, 87, 98, 125, 147, 192, 213, 271, 313, 389, 444, 551, 621, 767, 874, 1055, 1209, 1444, 1646, 1965, 2244, 2644, 2991

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OFFSET

0,6

COMMENTS

A 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.

A 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..40.

EXAMPLE

The a(4) = 1 through a(11) = 9 partitions:

211 221 21111 2221 422 22221 442 222221

2111 22111 22211 222111 4222 322211

211111 221111 2211111 222211 332111

2111111 21111111 322111 422111

2221111 2222111

22111111 3221111

211111111 22211111

221111111

2111111111

Conjugates of the a(4) = 1 through a(11) = 9 partitions:

(3,1) (3,2) (5,1) (4,3) (5,3) (5,4) (6,4) (6,5)

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

(6,1) (7,1) (7,2) (8,2) (8,3)

(3,3,1,1) (8,1) (9,1) (9,2)

(6,3,1) (10,1)

(3,3,2,2) (6,3,2)

(4,4,1,1) (6,4,1)

(7,3,1)

(6,3,1,1)

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.

These partitions are ranked by (A383516).

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).

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

A383508 counts partitions that are both Look-and-Say and section-sum (A383515).

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).

A383519 counts section-sum Wilf partitions (A383520).

Cf. A047966, A048767, A320348, A325324, A325325, A353837, A381431, A383511, A383530, A383709.

Sequence in context: A396030 A347494 A325324 * A383506 A241745 A332723

Adjacent sequences: A383506 A383507 A383508 * A383510 A383511 A383512

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, May 18 2025

STATUS

approved