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This shares with the permutation A122111 the property that each term of A028260 is mapped to a unique term of A244990 and each term of A026424 is mapped to a unique term of A244991.

This shares with the permutation A122111 the property that each term of A244990 is mapped to a unique term of A028260 and each term of A244991 is mapped to a unique term of A026424.

Factor n; replace each prime(i) with i, take the conjugate partition, replace parts i with prime(i) and multiply out.

From Antti Karttunen, May 12-19 2014: (Start)

For all n >= 1, A001222(a(n)) = A061395(n), and vice versa, A061395(a(n)) = A001222(n).

Because the partition conjugation doesn't change the partition's total sum, this permutation preserves A056239, i.e., A056239(a(n)) = A056239(n) for all n.

(Similarly, for all n, A001221(a(n)) = A001221(n), because the number of steps in the Ferrers/Young-diagram stays invariant under the conjugation. - Note added Apr 29 2022).

Because this permutation commutes with A241909, in other words, as a(A241909(n)) = A241909(a(n)) for all n, from which follows, because both permutations are self-inverse, that a(n) = A241909(a(A241909(n))), it means that this is also induced when partitions are conjugated in the partition enumeration system A241918. (Not only in A112798.)

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From Antti Karttunen, Jul 31 2014: (Start)

Rows in arrays A243060 and A243070 converge towards this sequence, and also, assuming no surprises at the rate of that convergence, this sequence occurs also as the central diagonal of both.

Each even number is mapped to a unique term of A102750 and vice versa.

Conversely, each odd number (larger than 1) is mapped to a unique term of A070003, and vice versa. The permutation pair A243287-A243288 has the same property. This is also used to induce the permutations A244981-A244984.

Taking the odd bisection and dividing out the largest prime factor results in the permutation A243505.

Shares with A245613 the property that each term of A028260 is mapped to a unique term of A244990 and each term of A026424 is mapped to a unique term of A244991.

Conversely, with A245614 (the inverse of above), shares the property that each term of A244990 is mapped to a unique term of A028260 and each term of A244991 is mapped to a unique term of A026424.

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The Maple program follows the steps described in the first comment. The subprogram C yields the conjugate partition of a given partition. - Emeric Deutsch, May 09 2015

The Heinz number of the partition that is conjugate to the partition with Heinz number n. The Heinz number of a partition p = [p_1, p_2, ..., p_r] is defined as Product(p_j-th prime, j=1...r). Example: a(3) = 4. Indeed, the partition with Heinz number 3 is [2]; its conjugate is [1,1] having Heinz number 4. - Emeric Deutsch, May 19 2015

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

For n > 0, also the number of partitions of n whose greatest part is even. [Edited by Gus Wiseman, Jan 05 2021]

Number of partitions of n+1 into an odd number of parts, the least being 1.

Also the number of partitions of n such that the number of even parts has the same parity as the number of odd parts; see Comments at A027193. - Clark Kimberling, Feb 01 2014, corrected Jan 06 2021

Suppose that c(0) = 1, that c(1), c(2), ... are indeterminates, that d(0) = 1, and that d(n) = -c(n) - c(n-1)*d(1) - ... - c(0)*d(n-1). When d(n) is expanded as a polynomial in c(1), c(2),..,c(n), the terms are of the form H*c(i_1)*c(i_2)*...*c(i_k). Let P(n) = [c(i_1), c(i_2), ..., c(i_k)], a partition of n. Then H is negative if P has an odd number of parts, and H is positive if P has an even number of parts. That is, d(n) has A027193(n) negative coefficients, A027187(n) positive coefficients, and A000041 terms. The maximal coefficient in d(n), in absolute value, is A102462(n). - Clark Kimberling, Dec 15 2016

Convolved with A036469 = A000070. - Gary W. Adamson, Jun 09 2009

Note that these partitions are located in the head of the last section of the set of partitions of n (see A135010). - Omar E. Pol, Nov 20 2009

Number of symmetric unimodal compositions of n+2 where the maximal part appears twice, see example. Also number of symmetric unimodal compositions of n where the maximal part appears an even number of times. - Joerg Arndt, Jun 11 2013

Number of partitions of n having parts of even multiplicity. These are the conjugates of the partitions from the definition. Example: a(8)=5 because we have [4,4],[3,3,1,1],[2,2,2,2],[2,2,1,1,1,1], and [1,1,1,1,1,1,1,1]. - Emeric Deutsch, Jan 27 2016

From Gus Wiseman, May 22 2021: (Start)

The Heinz numbers of the conjugate partitions described in Emeric Deutsch's comment above are given by A000290.

For n > 1, also the number of integer partitions of n-1 whose only odd part is the smallest. The Heinz numbers of these partitions are given by A341446. For example, the a(2) = 1 through a(14) = 15 partitions (empty columns shown as dots, A..D = 10..13) are:

1 . 3 . 5 . 7 . 9 . B . D

21 41 43 63 65 85

221 61 81 83 A3

421 441 A1 C1

2221 621 443 643

4221 641 661

22221 821 841

4421 A21

6221 4441

42221 6421

222221 8221

44221

62221

422221

2222221

Also the number of integer partitions of n whose greatest part is the sum of all the other parts. The Heinz numbers of these partitions are given by A344415. For example, the a(2) = 1 through a(12) = 11 partitions (empty columns not shown) are:

(11) (22) (33) (44) (55) (66)

(211) (321) (422) (532) (633)

(3111) (431) (541) (642)

(4211) (5221) (651)

(41111) (5311) (6222)

(52111) (6321)

(511111) (6411)

(62211)

(63111)

(621111)

(6111111)

Also the number of integer partitions of n of length n/2. The Heinz numbers of these partitions are given by A340387. For example, the a(2) = 1 through a(14) = 15 partitions (empty columns not shown) are:

(2) (22) (222) (2222) (22222) (222222) (2222222)

(31) (321) (3221) (32221) (322221) (3222221)

(411) (3311) (33211) (332211) (3322211)

(4211) (42211) (333111) (3332111)

(5111) (43111) (422211) (4222211)

(52111) (432111) (4322111)

(61111) (441111) (4331111)

(522111) (4421111)

(531111) (5222111)

(621111) (5321111)

(711111) (5411111)

(6221111)

(6311111)

(7211111)

(8111111)

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

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 partitions of n into distinct parts which are not powers of 2.

Also number of partitions of n into distinct parts such that the two largest parts differ by 1.

Also number of partitions of n such that the largest part occurs an odd number of times that is at least 3 and every other part occurs an even number of times. Example: a(10) = 2 because we have [2,2,2,1,1,1,1] and [2,2,2,2,2]. - Emeric Deutsch, Mar 30 2006

Also difference between number of partitions of 1+n into distinct parts and number of partitions of n into distinct parts. - Philippe LALLOUET, May 08 2007

In the Berndt reference replace {a -> -x, q -> x} in equation (3.1) to get f(x). G.f. is 1 - x * (1 - f(x)).

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Also number of symmetric unimodal compositions of n+3 where the maximal part appears three times. - Joerg Arndt, Jun 11 2013

Let c(n) = number of palindromic partitions of n whose greatest part has multiplicity 3; then c(n) = a(n-3) for n>=3. - Clark Kimberling, Mar 05 2014

From Gus Wiseman, Aug 22 2021: (Start)

Also the number of integer partitions of n - 1 whose parts cover an interval of positive integers starting with 2. These partitions are ranked by A339886. For example, the a(6) = 1 through a(16) = 5 partitions are:

32 222 322 332 432 3322 3332 4332 4432 5432 43332

2222 3222 22222 4322 33222 33322 33332 44322

32222 222222 43222 43322 333222

322222 332222 432222

2222222 3222222

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