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

Offset: 0

Author

Gus Wiseman, May 18 2025

Keywords

• nonn
• more

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.

Examples

			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)
		

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.

Programs

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}]