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A350839 Number of integer partitions of n with a difference < -1 and a conjugate difference < -1.

%I A350839 #8 Jan 29 2022 12:49:09
%S A350839 0,0,0,0,0,1,2,3,7,11,17,26,39,54,81,108,148,201,269,353,467,601,779,
%T A350839 995,1272,1605,2029,2538,3171,3941,4881,6012,7405,9058,11077,13478,
%U A350839 16373,19817,23953,28850,34692,41599,49802,59461,70905,84321,100155,118694
%N A350839 Number of integer partitions of n with a difference < -1 and a conjugate difference < -1.
%C A350839 We define a difference of a partition to be a difference of two adjacent parts.
%e A350839 The a(5) = 1 through a(10) = 17 partitions:
%e A350839   (311)  (411)   (511)    (422)     (522)      (622)
%e A350839          (3111)  (4111)   (611)     (711)      (811)
%e A350839                  (31111)  (3311)    (4221)     (4222)
%e A350839                           (4211)    (4311)     (4411)
%e A350839                           (5111)    (5211)     (5221)
%e A350839                           (41111)   (6111)     (5311)
%e A350839                           (311111)  (33111)    (6211)
%e A350839                                     (42111)    (7111)
%e A350839                                     (51111)    (42211)
%e A350839                                     (411111)   (43111)
%e A350839                                     (3111111)  (52111)
%e A350839                                                (61111)
%e A350839                                                (331111)
%e A350839                                                (421111)
%e A350839                                                (511111)
%e A350839                                                (4111111)
%e A350839                                                (31111111)
%t A350839 conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
%t A350839 Table[Length[Select[IntegerPartitions[n],(Min@@Differences[#]<-1)&&(Min@@Differences[conj[#]]<-1)&]],{n,0,30}]
%Y A350839 Allowing -1 gives A144300 = non-constant partitions.
%Y A350839 Taking one of the two conditions gives A239955, ranked by A073492, A065201.
%Y A350839 These partitions are ranked by A350841.
%Y A350839 A000041 = integer partitions, strict A000009.
%Y A350839 A034296 = flat (contiguous) partitions, strict A001227.
%Y A350839 A073491 = numbers whose prime indices have no gaps, strict A137793.
%Y A350839 A090858 = partitions with a single hole, ranked by A325284.
%Y A350839 A116931 = partitions with differences != -1, strict A003114.
%Y A350839 A116932 = partitions with differences != -1 or -2, strict A025157.
%Y A350839 A277103 = partitions with the same number of odd parts as their conjugate.
%Y A350839 A350837 = partitions with no adjacent doublings, strict A350840.
%Y A350839 A350842 = partitions with differences != -2, strict A350844, sets A005314.
%Y A350839 Cf. A000070, A000929, A008284, A040039, A183558, A319630, A321440, A350838.
%K A350839 nonn
%O A350839 0,7
%A A350839 _Gus Wiseman_, Jan 24 2022