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For a partition p, let l(p) = largest part of p, w(p) = number of 1's in p, m(p) = number of parts of p larger than w(p). The crank of p is given by l(p) if w(p) = 0, otherwise m(p)-w(p).

From Gus Wiseman, Mar 30 2021 and May 21 2022: (Start)

Also the number of even-length compositions of n with alternating parts strictly decreasing, or properly 2-colored partitions (proper = no equal parts of the same color) with the same number of parts of each color, or ordered pairs of strict partitions of the same length with total n. The odd-length case is A001522, and there are a total of A000041 compositions with alternating parts strictly decreasing (see A342528 for a bijective proof). The a(2) = 1 through a(7) = 8 ordered pairs of strict partitions of the same length are:

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

(2)(1) (2)(2) (2)(3) (2)(4) (2)(5)

(3)(1) (3)(2) (3)(3) (3)(4)

(4)(1) (4)(2) (4)(3)

(5)(1) (5)(2)

(21)(21) (6)(1)

(21)(31)

(31)(21)

Conjecture: Also the number of integer partitions y of n without a fixed point y(i) = i, ranked by A352826. This is stated at A238394, but Resta tells me he may not have had a proof. The a(2) = 1 through a(7) = 8 partitions without a fixed point are:

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

(21) (31) (41) (33) (43)

(211) (311) (51) (61)

(2111) (411) (331)

(3111) (511)

(21111) (4111)

(31111)

(211111)

The version for permutations is A000166, complement A002467.

The version for compositions is A238351.

This is column k = 0 of A352833.

A238352 counts reversed partitions by fixed points, rank statistic A352822.

A238394 counts reversed partitions without a fixed point, ranked by A352830.

A238395 counts reversed partitions with a fixed point, ranked by A352872. (End)

The above conjecture is true. See Section 4 of the Blecher-Knopfmacher paper in the Links section. - Jeremy Lovejoy, Sep 26 2022

Number of permutations p of {1,2,...,n} such that p(k)-k takes exactly two distinct values. Example: a(4)=4 because we have 4123, 3412, 2143 and 2341. - Max Alekseyev and Emeric Deutsch, Dec 22 2006

Number of solutions to the Diophantine equation xy + yz = n, with x,y,z >= 1.

In other words, number of ways to write n = (a + b) * k for positive integers a, b, k. - Gus Wiseman, Mar 25 2021

Not the same as A184396(n): a(66) = 136 while A184396(66) = 137. - Wesley Ivan Hurt, Dec 26 2013

From Gus Wiseman, Mar 25 2021: (Start)

Also the number of compositions of n into an even number of parts with alternating parts equal. These are finite even-length sequences q of positive integers summing to n such that q(i) = q(i+2) for all possible i. For example, the a(2) = 1 through a(8) = 11 compositions are:

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

(2,1) (2,2) (2,3) (2,4) (2,5) (2,6)

(3,1) (3,2) (3,3) (3,4) (3,5)

(1,1,1,1) (4,1) (4,2) (4,3) (4,4)

(5,1) (5,2) (5,3)

(1,2,1,2) (6,1) (6,2)

(2,1,2,1) (7,1)

(1,1,1,1,1,1) (1,3,1,3)

(2,2,2,2)

(3,1,3,1)

(1,1,1,1,1,1,1,1)

The odd-length version is A062968.

The version with alternating parts weakly decreasing is A114921, or A342528 if odd-length compositions are included.

The version with alternating parts unequal is A342532, or A224958 if odd-length compositions are included (unordered: A339404/A000726).

Allowing odd lengths as well as even gives A342527.

(End)

Inverse Möbius transform of n-1. - Wesley Ivan Hurt, Jun 29 2024

For a partition p, let l(p) = largest part of p, w(p) = number of 1's in p, m(p) = number of parts of p larger than w(p). The crank of p is given by l(p) if w(p) = 0, otherwise m(p)-w(p).

These are finite sequences q of positive integers summing to n such that q(i) >= q(i+2) for all possible i.

The strict case (alternating parts are strictly decreasing) is A000041. Is there a bijective proof?

Yes. Construct a Ferrers diagram by placing odd parts horizontally and even parts vertically in a fishbone pattern. The resulting Ferrers diagram will be for an ordinary partition and the process is reversible. It does not appear that this method can be applied to give a formula for this sequence. - Andrew Howroyd, Mar 25 2021

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

See A257989 or the program for a definition of crank of a partition.

These are finite sequences q of positive integers summing to n such that q(i) = q(i+2) for all possible i.

Old name was: Expansion of a q-series.

a(n) is also the number of 2-colored partitions of n with the same number of parts in each color. - Shishuo Fu, May 30 2017

From Gus Wiseman, Mar 25 2021: (Start)

Also the number of even-length compositions of n with alternating parts weakly decreasing. Allowing odd lengths also gives A342528. The version with alternating parts strictly decreasing appears to be A064428. The a(2) = 1 through a(7) = 16 compositions are:

(1,1) (1,2) (1,3) (1,4) (1,5) (1,6)

(2,1) (2,2) (2,3) (2,4) (2,5)

(3,1) (3,2) (3,3) (3,4)

(1,1,1,1) (4,1) (4,2) (4,3)

(1,2,1,1) (5,1) (5,2)

(2,1,1,1) (1,2,1,2) (6,1)

(1,3,1,1) (1,3,1,2)

(2,1,2,1) (1,4,1,1)

(2,2,1,1) (2,2,1,2)

(3,1,1,1) (2,2,2,1)

(1,1,1,1,1,1) (2,3,1,1)

(3,1,2,1)

(3,2,1,1)

(4,1,1,1)

(1,2,1,1,1,1)

(2,1,1,1,1,1)

(End)

The crank of a partition p is defined to be (i) the largest part of p if there is no 1 in p and (ii) (the number of parts larger than the number of 1's) minus (the number of 1's).

We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] as Product(p_j-th prime, j=1...r) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). For example, for the partition [1, 1, 2, 4, 10] we get 2*2*3*7*29 = 2436.

In the Maple program the subprogram B yields the partition with Heinz number n, the subprogram b yields the number of 1's in the partition with Heinz number n and the subprogram c yields the number of parts that are larger than the number of 1's in the partition with the Heinz number n.

These are finite even-length sequences q of positive integers summing to n such that q(i) != q(i+2) for all possible i.