cp's OEIS Frontend

A pseudo-logarithmic function in the sense that a(b*c) = a(b)+a(c) and so a(b^c) = c*a(b) and f(n) = k^a(n) is a multiplicative function. [Cf. A248692 for example.] Essentially a function from the positive integers onto the partitions of the nonnegative integers (1->0, 2->1, 3->2, 4->1+1, 5->3, 6->1+2, etc.) so each value a(n) appears A000041(a(n)) times, first with the a(n)-th prime and last with the a(n)-th power of 2. Produces triangular numbers from primorials. - Henry Bottomley, Nov 22 2001

Michael Nyvang writes (May 08 2006) that the Danish composer Karl Aage Rasmussen discovered this sequence in the 1990's: it has excellent musical properties.

All A000041(a(n)) different n's with the same value a(n) are listed in row a(n) of triangle A215366. - Alois P. Heinz, Aug 09 2012

a(n) is the sum of the parts of the partition having Heinz number n. We define the Heinz number of a partition p = [p_1, p_2, ..., p_r] as Product_{j=1..r} (p_j-th prime) (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. Example: a(33) = 7 because the partition with Heinz number 33 = 3 * 11 is [2,5]. - Emeric Deutsch, May 19 2015

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.)

(End)

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.

(End)

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

Also number of partitions of n with positive crank (n>=2), cf. A064391. - Vladeta Jovovic, Sep 30 2001

Number of smooth weakly unimodal compositions of n into positive parts such that the first and last part are 1 (smooth means that successive parts differ by at most one), see example. Dropping the requirement for unimodality gives A186085. - Joerg Arndt, Dec 09 2012

Number of weakly unimodal compositions of n where the maximal part m appears at least m times, see example. - Joerg Arndt, Jun 11 2013

Also weakly unimodal compositions of n with first part 1, maximal up-step 1, and no consecutive up-steps; see example. The smooth weakly unimodal compositions are recovered by shifting all rows above the bottom row to the left by one position with respect to the next lower row. - Joerg Arndt, Mar 30 2014

It would seem from Stanley that he regards a(0)=0 for this sequence and A001523. - Michael Somos, Feb 22 2015

From Gus Wiseman, Mar 30 2021: (Start)

Also the number of odd-length compositions of n with alternating parts strictly decreasing. These are finite odd-length sequences q of positive integers summing to n such that q(i) > q(i+2) for all possible i. The even-length version is A064428. For example, the a(1) = 1 through a(9) = 14 compositions are:

(1) (2) (3) (4) (5) (6) (7) (8) (9)

(211) (221) (231) (241) (251) (261)

(311) (312) (322) (332) (342)

(321) (331) (341) (351)

(411) (412) (413) (423)

(421) (422) (432)

(511) (431) (441)

(512) (513)

(521) (522)

(611) (531)

(612)

(621)

(711)

(32211)

(End)

In the Ferrers diagram of a partition x of n, count the dots in each diagonal parallel to the main diagonal (starting at the top-right, say). The result diag(x) is a smooth weakly unimodal composition of n into positive parts such that the first and last part are 1. For example, diag(5541) = 11233221. The function diag is many-to-one; the size of its codomain as a set is a(n). If diag(x) = diag(y), each hook of x can be slid by the same amount past the main diagonal to get y. For example, diag(5541) = diag(44331). - George Beck, Sep 26 2021

From Gus Wiseman, May 23 2022: (Start)

Conjecture: Also the number of integer partitions y of n with a fixed point y(i) = i. These partitions are ranked by A352827. The conjecture is stated at A238395, but Resta tells me he may not have had a proof. The a(1) = 1 through a(8) = 10 partitions are:

(1) (11) (111) (22) (32) (42) (52) (62)

(1111) (221) (222) (322) (422)

(11111) (321) (421) (521)

(2211) (2221) (2222)

(111111) (3211) (3221)

(22111) (4211)

(1111111) (22211)

(32111)

(221111)

(11111111)

Note that these are not the same partitions (compare A352827 to A352874), only the same count (apparently).

(End)

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

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)

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

From Gus Wiseman, May 21 2022: (Start)

Also the number of integer partitions of n with k parts below the diagonal. For example, the partition (3,2,2,1) has two parts (at positions 3 and 4) below the diagonal (1,2,3,4). Row n = 8 counts the following partitions:

8 71 611 5111 41111 311111 2111111 11111111

44 332 2222 22211 221111

53 422 3221 32111

62 431 3311

521 4211

Indices of parts below the diagonal are also called strong nonexcedances.

(End)

Also number of partitions of n with even number of even parts. There is no restriction on the odd parts.

a(n) = u(n) + v(n), n >= 2, of the Osima reference, p. 383.

Also number of partitions of n with largest part congruent to n modulo 2: a(2*n) = A027187(2*n), a(2*n-1) = A027193(2*n-1); a(n) = A000041(n) - A000701(n). - Reinhard Zumkeller, Apr 22 2006

Equivalently, number of partitions of n with number of parts having the same parity as n. - Olivier Gérard, Apr 04 2012

Also number of distinct free Young diagrams (Ferrers graphs with n nodes). Free Young diagrams are distinct when none is a rigid transformation (translation, rotation, reflection or glide reflection) of another. - Jani Melik, May 08 2016

Let the cycle type of an even permutation be represented by the partition A=(O1,O2,...,Oi,E1,E2,...,E2j), where the Os are parts with odd length and the Es are parts with even lengths, and where j may be zero, using Reinhard Zumkeller's observation that the partition associated with a cycle type of an even permutation has an even number of even parts. The set of even cycle types enumerated here can be considered a monoid under a binary operation *: Let A be as above and B=(o1,o2,...,ok,e1,e2,...,e2m). A*B is the partition (O1o1,O1o2,...,O1ok,O1e1,...,O1e2m,O2o1,...,O2e2m,...,Oio1,...,Oie2m,E1o1,...,E1e2m,...,E2je2m). This product has 2im+2jk+4jm even parts, so it represents the cycle type of an even permutation. - Richard Locke Peterson, Aug 20 2018

From Gus Wiseman, Mar 31 2022: (Start)

Also the number of integer partitions of n with Heinz number greater than or equal to 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 A352488. The complement is counted by A000701. For example, the a(n) partitions for n = 1...7 are:

(1) (11) (21) (22) (221) (222) (331)

(111) (211) (311) (321) (2221)

(1111) (2111) (2211) (3211)

(11111) (3111) (4111)

(21111) (22111)

(111111) (31111)

(211111)

(1111111)

Also the number of integer partitions of n with Heinz number less than or equal to their conjugate, ranked by A352489. For example, the a(n) partitions for n = 1...7 are:

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

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

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

(311) (51) (61)

(321) (322)

(411) (421)

(511)

(4111)

(End)

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

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.

Also numbers m whose prime indices have a submultiset summing to bigomega(m).

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

These are the Heinz numbers of the partitions counted by A367212.