cp's OEIS Frontend

A383508 Number of integer partitions of n that are both Look-and-Say and section-sum partitions.

%I A383508 #13 May 18 2025 14:37:34
%S A383508 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,
%T A383508 174,199,225,263,299,337,388,435,488,545,635,681,775,841,948,1051,
%U A383508 1181,1271,1446,1553,1765,1896,2141,2285,2608,2799
%N A383508 Number of integer partitions of n that are both Look-and-Say and section-sum partitions.
%C A383508 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.
%C A383508 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.
%e A383508 The a(1) = 1 through a(8) = 9 partitions:
%e A383508   (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
%e A383508        (11)  (111)  (22)    (311)    (33)      (322)      (44)
%e A383508                     (1111)  (11111)  (222)     (331)      (332)
%e A383508                                      (411)     (511)      (611)
%e A383508                                      (3111)    (4111)     (2222)
%e A383508                                      (111111)  (31111)    (5111)
%e A383508                                                (1111111)  (41111)
%e A383508                                                           (311111)
%e A383508                                                           (11111111)
%t A383508 disjointFamilies[y_]:=Select[Tuples[IntegerPartitions /@ Length/@Split[y]],UnsameQ@@Join@@#&];
%t A383508 conj[y_]:=If[Length[y]==0,y, Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
%t A383508 Table[Length[Select[IntegerPartitions[n], disjointFamilies[#]!={}&&disjointFamilies[conj[#]]!={}&]], {n,0,30}]
%Y A383508 Ranking sequences are shown in parentheses below.
%Y A383508 The non Wilf case is A383511 (A383518).
%Y A383508 These partitions are ranked by (A383515).
%Y A383508 A000041 counts integer partitions, strict A000009.
%Y A383508 A047993 counts partitions with max part = length (A106529).
%Y A383508 A098859 counts Wilf partitions (A130091), conjugate (A383512).
%Y A383508 A122111 represents conjugation in terms of Heinz numbers.
%Y A383508 A239455 counts Look-and-Say partitions (A351294), complement A351293 (A351295).
%Y A383508 A239455 counts section-sum partitions (A381432), complement A351293 (A381433).
%Y A383508 A336866 counts non Wilf partitions (A130092), conjugate (A383513).
%Y A383508 A351592 counts non Wilf Look-and-Say partitions (A384006).
%Y A383508 A383509 counts partitions that are Look-and-Say but not section-sum (A383516).
%Y A383508 A383509 counts partitions that are not Look-and-Say but are section-sum (A384007).
%Y A383508 A383510 counts partitions that are neither Look-and-Say nor section-sum (A383517).
%Y A383508 Cf. A033461, A047966, A048767, A320348, A325324, A325325, A381431, A383519, A383530, A383709.
%K A383508 nonn
%O A383508 0,3
%A A383508 _Gus Wiseman_, May 17 2025