A239955 Number of partitions p of n such that (number of distinct parts of p) <= max(p) - min(p).

0, 0, 0, 0, 1, 2, 4, 7, 12, 17, 27, 38, 54, 75, 104, 137, 187, 245, 322, 418, 542, 691, 887, 1121, 1417, 1777, 2228, 2767, 3441, 4247, 5235, 6424, 7871, 9594, 11688, 14173, 17168, 20723, 24979, 30008, 36010, 43085, 51479, 61357, 73032, 86718, 102852, 121718

Offset: 0

Author

Clark Kimberling, Mar 30 2014

Keywords

• nonn
• easy
• changed

Comments

From Gus Wiseman, Jun 26 2022: (Start)

Also the number of partitions of n with at least one gap, i.e., partitions whose parts do not form a contiguous interval. These partitions are ranked by A073492. For example, the a(0) = 0 through a(8) = 12 partitions are:

. . . . (31) (41) (42) (52) (53)

(311) (51) (61) (62)

(411) (331) (71)

(3111) (421) (422)

(511) (431)

(4111) (521)

(31111) (611)

(3311)

(4211)

(5111)

(41111)

(311111)

Also the number of non-constant partitions of n with a repeated non-maximal part, ranked by A065201. The a(0) = 0 through a(8) = 12 partitions are:

. . . . (211) (311) (411) (322) (422)

(2111) (2211) (511) (611)

(3111) (3211) (3221)

(21111) (4111) (3311)

(22111) (4211)

(31111) (5111)

(211111) (22211)

(32111)

(41111)

(221111)

(311111)

(2111111)

(End)

Examples

			a(6) counts these 4 partitions:  51, 42, 411, 3111.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 201 terms from John Tyler Rascoe)

Crossrefs

Cf. A239954, A239956, A239958.

The complement is counted by A034296 (strict A137793), ranked by A073491.

These partitions are ranked by A073492, conjugate A065201.

Applying the condition to the conjugate gives A350839, ranked by A350841.

A000041 counts integer partitions, strict A000009.

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

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

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

Cf. A000070, A001227, A008284, A144300, A183558, A321440.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i<1, 0, add(b(n-i*j, i-1), j=1..n/i)))
    end:
a:= n-> combinat[numbpart](n)-add(b(n, k), k=0..n):
seq(a(n), n=0..47);  # Alois P. Heinz, Aug 18 2025

Mathematica

z = 60; d[p_] := d[p] = Length[DeleteDuplicates[p]]; f[p_] := f[p] = Max[p] - Min[p]; g[n_] := g[n] = IntegerPartitions[n];
Table[Count[g[n], p_ /; d[p] < f[p]], {n, 0, z}]  (*A239954*)
Table[Count[g[n], p_ /; d[p] <= f[p]], {n, 0, z}] (*A239955*)
Table[Count[g[n], p_ /; d[p] == f[p]], {n, 0, z}] (*A239956*)
Table[Count[g[n], p_ /; d[p] > f[p]], {n, 0, z}]  (*A034296*)
Table[Count[g[n], p_ /; d[p] >= f[p]], {n, 0, z}] (*A239958*)
(* second program *)
Table[Length[Select[IntegerPartitions[n],Min@@Differences[#]<-1&]],{n,0,30}] (* Gus Wiseman, Jun 26 2022 *)

PARI

qs(a,q,n) = {prod(k=0,n,1-a*q^k)}
A_q(N) = {if(N<4, vector(N+1,i,0), my(q='q+O('q^(N-2)), g= sum(i=2,N+1, q^i/qs(q,q,i-1)*sum(j=1,i-1, q^(2*j)*qs(q^2,q^2,j-2)))); concat([0,0,0,0], Vec(g)))} \\ John Tyler Rascoe, Aug 16 2025

Formula

a(n) = A000041(n) - A034296(n).

G.f.: Sum_{i>1} q^i/(q;q){i-1} * Sum{j=1..i-1} (q^2;q^2){j-2} where (a;q)_k = Product{i>=0..k} (1-a*q^i). - John Tyler Rascoe, Aug 16 2025