cp's OEIS Frontend

Also number of partitions of 2*n with exactly 2 odd parts (offset 1). - Vladeta Jovovic, Jan 12 2005

Also number of transitions from one partition of n+2 to another, where a transition consists of replacing any two parts with their sum. Remove all 1' and 2' from the partition, replacing them with ((number of 2') + 1) and ((number of 1') + (number of 2') + 1); these are the two parts being summed. Number of partitions of n into parts of 2 kinds with at most 2 parts of the second kind, or of n+2 into parts of 2 kinds with exactly 2 parts of the second kind. - Franklin T. Adams-Watters, Mar 20 2006

From Christian Gutschwager (gutschwager(AT)math.uni-hannover.de), Feb 10 2010: (Start)

a(n) is also the number of pairs of partitions of n+2 which differ by only one box (for bijection see the first Gutschwager link).

a(n) is also the number of partitions of n with two parts marked.

a(n) is also the number of partitions of n+1 with two different parts marked. (End)

Convolution of A000041 and A008619. - Vaclav Kotesovec, Aug 18 2015

a(n) = P(/2,n), a particular case of P(/k,n) defined as follows: P(/0,n) = A000041(n) and P(/k,n) = P(/k-1, n) + P(/k-1,n-k) + P(/k-1, n-2k) + ... Also, P(/k,n) = the convolution of A000041 and the partitions of n with exactly k parts, and g.f. P(/k,n) = (g.f. for P(n)) * 1/(1-x)...(1-x^k). - Gregory L. Simay, Mar 22 2018

a(n) is also the sum of binomial(D(p),2) in partitions p of (n+3), where D(p)= number of different sizes of parts in p. - Emily Anible, Apr 03 2018

Also partitions of 2*(n+1) with alternating sum 2. Also partitions of 2*(n+1) with reverse-alternating sum -2 or 2. - Gus Wiseman, Jun 21 2021

Define the distance graph of the partitions of n using the distance function in A366156 as follows: two vertices (partitions) share an edge if and only if the distance between the vertices is 2. Then a(n) is the number of edges in the distance graph of the partitions of n. - Clark Kimberling, Oct 12 2023

The reverse-alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i.

Also the number of reversed integer partitions of n with alternating sum >= 0.

A formula for the reverse-alternating sum of a partition is: (-1)^(k-1) times the number of odd parts in the conjugate partition, where k is the number of parts. So a(n) is the number of integer partitions of n whose conjugate parts are all even or whose length is odd. By conjugation, this is also the number of integer partitions of n whose parts are all even or whose greatest part is odd.

All integer partitions have alternating sum >= 0, so the non-reversed version is A000041.

Is this sequence weakly increasing? In particular, is A344611(n) <= A160786(n)?

The trivariate g.f. with x marking weight (i.e., sum of the parts), t marking number of odd parts and s marking number of even parts, is 1/product((1-tx^(2j-1))(1-sx^(2j)), j=1..infinity). - Emeric Deutsch, Mar 30 2006

The alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. This is equal to the number of odd parts in the conjugate partition, so T(n,k) is the number of integer partitions of n with k odd parts in the conjugate partition, which is also the number of partitions of n with k odd parts.

Also the number of integer partitions of n with odd-indexed parts (odd bisection) summing to k, ceiling(n/2) <= k <= n. The even-indexed version is A346633. - Gus Wiseman, Nov 29 2021

Ramanujan theta functions: phi(q) (A000122), chi(q) (A000700).

The reverse-alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i. This is equal to (-1)^(k-1) times the number of odd parts in the conjugate partition, where k is the number of parts.

Also the number of reversed integer partitions of 2n with alternating sum 2k.

The partitions of n are partitioned into four types:

EO, even # of odd parts and odd # of even parts, A236559;

OE, odd # of odd parts and even # of even parts, A160786;

EE, even # of odd parts and even # of even parts, A236913;

OO, odd # of odd parts and odd # of even parts, A236914.

A236559 and A160786 are the bisections of A027193;

A236913 and A236914 are the bisections of A027187.

Also number of cycle types of odd permutations.

Also number of partitions of n with an odd number of even parts. There is no restriction on the odd parts. - N. Sato, Jul 20 2005. E.g., a(6)=5 because we have [6],[4,1,1],[3,2,1],[2,2,2] and [2,1,1,1,1]. - Emeric Deutsch, Mar 02 2006

Also number of partitions of n with largest part not congruent to n modulo 2: a(2*n)=A027193(2*n), a(2*n+1)=A027187(2*n+1); a(n)=A000041(n)-A046682(n). - Reinhard Zumkeller, Apr 22 2006

From Gus Wiseman, Mar 31 2022: (Start)

Also the number of integer partitions of n with Heinz number greater than that of their conjugate, where the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). These partitions are ranked by A352490. The complement is counted by A046682. For example, the a(n) partitions for n = 2...8 are:

(11) (111) (211) (221) (222) (331) (2222)

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

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

(21111) (22111) (22211)

(111111) (31111) (32111)

(211111) (41111)

(1111111) (221111)

(311111)

(2111111)

(11111111)

Also the number of integer partitions of n with Heinz number less than that of their conjugate, ranked by A352487. For example, the a(n) partitions for n = 2...8 are:

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

(31) (32) (33) (43) (44)

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

(51) (61) (62)

(411) (322) (71)

(421) (422)

(511) (431)

(521)

(611)

(5111)

(End)

The bisection {1,1,2,3,5,7,11,15,22,...} agrees with the initial terms of A008641, Number of partitions of n into at most 12 parts and also A008635, Molien series for A_12.

a(2n+1)=a(2n) for all n>0. If the partition {...,1} is a member of a(2n) then the partition {...,1,1} is a member of a(2n+1). - Robert G. Wilson v, Jun 09 2006

Number of partitions of n where all parts (except for possibly the first part) are even; see example. - Joerg Arndt, Apr 22 2013

For n >= 2, a(n) = number of partitions p of n such that floor(n/2) is a part of p. For n >= 1, a(n) = number of partitions p of n such that ceiling(n/2) is a part of p. - Clark Kimberling, Feb 28 2014

From Gus Wiseman, Oct 28 2021: (Start)

If we insert zeros every three terms, this counts partitions of n such that n = floor(3*k/2), where k is the number of parts. This counts by sum rather than length. These partitions are ranked by A347452.

Also the number of integer partitions of n with alternating product 1, where the alternating product of a sequence (y_1,...,y_k) is Product_i y_i^((-1)^(i-1)). These are the conjugates of the partitions (ranked by A336119) described in Arndt's comment above. For example, the a(2) = 1 through a(10) = 7 partitions are:

11 111 22 221 33 331 44 441 55

1111 11111 2211 22111 2222 22221 3322

111111 1111111 3311 33111 4411

221111 2211111 222211

11111111 111111111 331111

22111111

1111111111

These partitions are ranked by A028982. The odd-length case is A035363 (shifted), which is also the version for sum instead of product. The multiplicative version (factorizations) is A347438.

(End)

The reverse-alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i.

Also the number of reversed integer partitions of 2n with alternating sum >= 0.

The reverse-alternating sum of a partition is equal to (-1)^(k-1) times the number of odd parts in the conjugate partition, where k is the number of parts. So a(n) is the number of partitions of 2n whose conjugate parts are all even or whose length is odd. By conjugation, this is also the number of partitions of 2n whose parts are all even or whose greatest part is odd.