A064174 Number of partitions of n with nonnegative rank.

1, 1, 2, 3, 4, 6, 9, 12, 17, 23, 31, 42, 56, 73, 96, 125, 161, 207, 265, 336, 426, 536, 672, 840, 1046, 1296, 1603, 1975, 2425, 2970, 3628, 4417, 5367, 6503, 7861, 9482, 11412, 13702, 16423, 19642, 23447, 27938, 33231, 39453, 46767, 55342, 65386, 77135

Offset

1,3

Comments

The rank of a partition is the largest summand minus the number of summands.

This sequence (up to proof) equals "partitions of 2n with even number of parts, ending in 1, with max descent of 1, where the number of odd parts in odd places equals the number of odd parts in even places. (See link and 2nd Mathematica line.) - Wouter Meeussen, Mar 29 2013

Number of partitions p of n such that max(max(p), number of parts of p) is a part of p. - Clark Kimberling, Feb 28 2014

From Gus Wiseman, Mar 09 2019: (Start)

Also the number of integer partitions of n with maximum part greater than or equal to the number of parts. The Heinz numbers of these integer partitions are given by A324521. For example, the a(1) = 1 through a(8) = 12 partitions are:

(1) (2) (3) (4) (5) (6) (7) (8)

(21) (22) (32) (33) (43) (44)

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

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

(321) (322) (71)

(411) (331) (332)

(421) (422)

(511) (431)

(4111) (521)

(611)

(4211)

(5111)

Also the number of integer partitions of n with maximum part less than or equal to the number of parts. The Heinz numbers of these integer partitions are given by A324562. For example, the a(1) = 1 through a(8) = 12 partitions are:

(1) (11) (21) (22) (221) (222) (322) (332)

(111) (211) (311) (321) (331) (2222)

(1111) (2111) (2211) (2221) (3221)

(11111) (3111) (3211) (3311)

(21111) (4111) (4211)

(111111) (22111) (22211)

(31111) (32111)

(211111) (41111)

(1111111) (221111)

(311111)

(2111111)

(11111111)

(End)

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Alois P. Heinz)

Cristina Ballantine and Mircea Merca, Bisected theta series, least r-gaps in partitions, and polygonal numbers, arXiv:1710.05960 [math.CO], 2017.

Rekha Biswal, bijection between number of partitions of 2n satisfying certain conditions with number of partitions of n, Mathoverflow.

Mircea Merca, Rank partition functions and truncated theta identities, arXiv:2006.07705 [math.CO], 2020.

Formula

a(n) = (A000041(n) + A047993(n))/2.

a(n) = p(n-1) - p(n-5) + p(n-12) - ... -(-1)^k*p(n-(3*k^2-k)/2) + ..., where p() is A000041(). - Vladeta Jovovic, Aug 04 2004

G.f.: Sum_{n>=1} x^n * Product_{k=1..n} (1 - x^(n+k-1))/(1 - x^k). - Paul D. Hanna, Aug 03 2015

A064173(n) + a(n) = A000041(n). - R. J. Mathar, Feb 22 2023

G.f.: (1/Product_{k>=1} (1-x^k)) * Sum_{k>=1} (-1)^(k-1) * x^(k*(3*k-1)/2). - Seiichi Manyama, May 21 2023

Example

a(20) = p(19) - p(15) + p(8) = 490 - 176 + 22 = 336.

Maple

f:= n -> add((-1)^(k+1)*combinat:-numbpart(n-(3*k^2-k)/2), k=1..floor((1+sqrt(24*n+1))/6)):
map(f, [$1..100]); # Robert Israel, Aug 03 2015

Mathematica

Table[Count[IntegerPartitions[n], q_ /; First[q] >= Length[q]], {n, 16}]
(* also *)
Table[Count[IntegerPartitions[2n], q_/; Last[q]===1 && Max[q-PadRight[Rest[q], Length[q]]]<=1 && Count[First/@Partition[q, 2], _?OddQ]==Count[Last/@Partition[q, 2], _?OddQ]], {n, 16}]
(* also *)
Table[Count[IntegerPartitions[n], p_ /; MemberQ[p, Max[Max[p], Length[p]]]], {n, 50}] (* Clark Kimberling, Feb 28 2014 *)

Prog

(PARI) {a(n) = my(A=1); A = sum(m=0, n, x^m*prod(k=1, m, (1-x^(m+k-1))/(1-x^k +x*O(x^n)))); polcoeff(A, n)}
for(n=1, 60, print1(a(n), ", ")) \\ Paul D. Hanna, Aug 03 2015
(PARI) my(N=50, x='x+O('x^N)); Vec(1/prod(k=1, N, 1-x^k)*sum(k=1, N, (-1)^(k-1)*x^(k*(3*k-1)/2))) \\ Seiichi Manyama, May 21 2023

Crossrefs

Cf. A063995, A064173 (complement).

Row sums of triangle A105806.

Cf. A003114, A006141, A039900, A047993, A090858, A106529, A133121.

Cf. A324516, A324518, A324520, A324521, A324522, A324560, A324562, A324572.

Sequence in context: A035992 A036003 A027338 * A062121 A094995 A018591

Adjacent sequences: A064171 A064172 A064173 * A064175 A064176 A064177

Keyword

nonn

Author

Vladeta Jovovic, Sep 20 2001

Extensions

Mathematica programs modified by Clark Kimberling, Feb 12 2014

Status

approved