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

0, 0, 0, 0, 0, 1, 2, 3, 7, 11, 17, 26, 39, 54, 81, 108, 148, 201, 269, 353, 467, 601, 779, 995, 1272, 1605, 2029, 2538, 3171, 3941, 4881, 6012, 7405, 9058, 11077, 13478, 16373, 19817, 23953, 28850, 34692, 41599, 49802, 59461, 70905, 84321, 100155, 118694

Offset: 0

Author

Gus Wiseman, Jan 24 2022

Keywords

• nonn

Comments

We define a difference of a partition to be a difference of two adjacent parts.

Examples

			The a(5) = 1 through a(10) = 17 partitions:
  (311)  (411)   (511)    (422)     (522)      (622)
         (3111)  (4111)   (611)     (711)      (811)
                 (31111)  (3311)    (4221)     (4222)
                          (4211)    (4311)     (4411)
                          (5111)    (5211)     (5221)
                          (41111)   (6111)     (5311)
                          (311111)  (33111)    (6211)
                                    (42111)    (7111)
                                    (51111)    (42211)
                                    (411111)   (43111)
                                    (3111111)  (52111)
                                               (61111)
                                               (331111)
                                               (421111)
                                               (511111)
                                               (4111111)
                                               (31111111)
		

Crossrefs

Allowing -1 gives A144300 = non-constant partitions.

Taking one of the two conditions gives A239955, ranked by A073492, A065201.

These partitions are ranked by A350841.

A000041 = integer partitions, strict A000009.

A034296 = flat (contiguous) partitions, strict A001227.

A073491 = numbers whose prime indices have no gaps, strict A137793.

A090858 = partitions with a single hole, ranked by A325284.

A116931 = partitions with differences != -1, strict A003114.

A116932 = partitions with differences != -1 or -2, strict A025157.

A277103 = partitions with the same number of odd parts as their conjugate.

A350837 = partitions with no adjacent doublings, strict A350840.

A350842 = partitions with differences != -2, strict A350844, sets A005314.

Cf. A000070, A000929, A008284, A040039, A183558, A319630, A321440, A350838.

Programs

Mathematica

conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[Length[Select[IntegerPartitions[n],(Min@@Differences[#]<-1)&&(Min@@Differences[conj[#]]<-1)&]],{n,0,30}]