A383511 Number of integer partitions of n that are Look-and-Say and section-sum but not Wilf.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 1, 0, 1, 0, 3, 3, 0, 0, 5, 2, 1, 5, 6, 1, 10, 5, 12, 11, 12, 14, 31, 15, 25, 28, 38

Offset

0,16

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.

A 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(n) partitions for n = 12, 15, 20, 24, 28:
  (6,3,3)  (6,6,3)    (8,8,4)    (12,6,6)         (14,7,7)
           (6,3,3,3)  (10,5,5)   (6,6,6,3,3)      (8,8,8,4)
                      (8,4,4,4)  (8,4,4,4,4)      (8,8,4,4,4)
                                 (6,6,3,3,3,3)    (8,4,4,4,4,4)
                                 (6,3,3,3,3,3,3)  (10,6,6,2,2,2)
                                                  (11,6,6,1,1,1,1,1)

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[#]]!={} && !UnsameQ@@Length/@Split[#]&]], {n, 0, 30}]

Crossrefs

Ranking sequences are shown in parentheses below.

This is the non Wilf case of A383508 (A383515).

These partitions are ranked by (A383518).

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).

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).

A383519 counts section-sum Wilf partitions (A383520).

Cf. A033461, A047966, A048767, A122111, A217605, A320348, A325324, A325325, A381431, A383530, A383709.

Sequence in context: A123022 A072943 A372596 * A072175 A280712 A363926

Adjacent sequences: A383508 A383509 A383510 * A383512 A383513 A383514

Keyword

nonn,more

Author

Gus Wiseman, May 18 2025

Status

approved