cp's OEIS Frontend

Also the total number of all different integers in all partitions of n + 1. E.g., a(3) = 7 because the partitions of 4 comprise the sets {1},{1, 2},{2},{1, 3},{4} of different integers and their total number is 7. - Thomas Wieder, Apr 10 2004

With offset 1, also the number of 1's in all partitions of n. For example, 3 = 2+1 = 1+1+1, a(3) = (zero 1's) + (one 1's) + (three 1's), so a(3) = 4. - Naohiro Nomoto, Jan 09 2002. See the Riordan reference p. 184, last formula, first term, for a proof based on Fine's identity given in Riordan, p. 182 (20).

Also, number of partitions of n into parts when there are two kinds of parts of size one.

Also number of graphical forest partitions of 2n+2.

a(n) = count 2 for each partition of n and 1 for each decrement. E.g., the partitions of 4 are 4 (2), 31 (3), 22 (2), 211 (3) and 1111 (2). 2 + 3 + 2 + 3 + 2 = 12. This is related to the Ferrers representation. We can see that taking the Ferrers diagram for each partition of n and adding a new * to all available columns, we generate each partition of n+1, but with repeats (A058884). - Jon Perry, Feb 06 2004

Also the number of 1-transitions among all integer partitions of n. A 1-transition is the removal of a digit "1" from a partition containing at least one "1" and subsequent addition of that "1" to another digit in that partition. This other digit may be a "1" also, but all digits of equal amount are considered as undistinquishable (unlabeled). E.g., for n=6 one has the partition [1113] for which the following two 1-transitions are possible: [1113] --> [123] and [1113] --> [114]. The 1-transitions of n form a partial order (poset). For n=6 one has 12 1-transitions: [111111] --> [11112], [11112] --> [1113], [11112] --> [1122], [1113] --> [114], [1113] --> [123], [1122] --> [123], [1122] --> [222], [123] --> [33], [123] --> [24], [114] --> [15], [114] --> [24], [15] --> [6]. - Thomas Wieder, Mar 08 2005

Also number of partitions of 2n+1 where one of the parts is greater than n (also where there are more than n parts) and of 2n+2 where one of the parts is greater than n+1 (or with more than n+1 parts). - Henry Bottomley, Aug 01 2005

Equals left border of triangle A137633 - Gary W. Adamson, Jan 31 2008

Equals row sums of triangle A027293. - Gary W. Adamson, Oct 26 2008

Convolved with A010815 = [1,1,1,...]. n-th partial sum of A000041 convolved with A010815 = the binomial sequence starting (1, n, ...). - Gary W. Adamson, Nov 09 2008

Equals A036469 convolved with A035363. - Gary W. Adamson, Jun 09 2009

a(A004526(n)) = A025065(n). - Reinhard Zumkeller, Jan 23 2010

a(n) = if n <= 1 then A054225(1,n) else A054225(n,1). - Reinhard Zumkeller, Nov 30 2011

Also the total number of 1's among all hook-lengths in all partitions of n. E.g., a(4)=7 because hooks of the partitions of n = 4 comprise the multisets {4,3,2,1}, {4,2,1,1}, {3,2,2,1}, {4,1,2,1}, {4,3,2,1} and their total number of 1's is 7. - T. Amdeberhan, Jun 03 2012

With offset 1, a(n) is also the difference between the sum of largest and the sum of second largest elements in all partitions of n. More generally, the number of occurrences of k in all partitions of n equals the difference between the sum of k-th largest and the sum of (k+1)st largest elements in all partitions of n. And more generally, the sum of the number of occurrences of k, k+1, k+2..k+m in all partitions of n equals the difference between the sum of k-th largest and the sum of (k+m+1)st largest elements in all partitions of n. - Omar E. Pol, Oct 25 2012

a(0) = 1 and 2*a(n-1) >= a(n) for all n > 0. Hence a(n) is a complete sequence. - Frank M Jackson, Apr 08 2013

a(n) is the number of conjugacy classes in the order-preserving, order-decreasing and (order-preserving and order-decreasing) injective transformation semigroups. - Ugbene Ifeanyichukwu, Jun 03 2015

a(n) is also the number of unlabeled subgraphs of the n-cycle C_n. For example, for n = 3, there are 3 unlabeled subgraphs of the triangle C_3 with 0 edges, 2 with 1 edge, 1 with 2 edges, and 1 with 3 edges (C_3 itself), so a(3) = 3 + 2 + 1 + 1 = 7. - John P. McSorley, Nov 21 2016

a(n) is also the number of partitions of 2n with all parts either even or equal to 1. Proof: the number of such partitions of 2n with exactly 2k 1's is p(n-k), for k = 0,..,n. Summing over k gives the formula. - Leonard Chastkofsky, Jul 24 2018

a(n) is the total number of polygamma functions that appear in the expansion of the (n+1)st derivative of x! with respect to x. More specifically, a(n) is the number of times the string "PolyGamma" appears in the expansion of D[x!, {x, n + 1}] in Mathematica. For example, D[x!, {x, 3 + 1}] = Gamma[1 + x] PolyGamma[0, 1 + x]^4 + 6 Gamma[1 + x] PolyGamma[0, 1 + x]^2 PolyGamma[1, 1 + x] + 3 Gamma[1 + x] PolyGamma[1, 1 + x]^2 + 4 Gamma[1 + x] PolyGamma[0, 1 + x] PolyGamma[2, 1 + x] + Gamma[1 + x] PolyGamma[3, 1 + x], and we see that the string "PolyGamma" appears a total of a(3) = 7 times in this expansion. - John M. Campbell, Aug 11 2018

With offset 1, also the number of integer partitions of 2n that do not comprise the multiset of vertex-degrees of any multigraph (i.e., non-multigraphical partitions); see A209816 for multigraphical partitions. - Gus Wiseman, Oct 26 2018

Also a(n) is the number of partitions of 2n+1 with exactly one odd part.

Delete the odd part 2k+1, k=0, ..., n, to get a partition of 2n-2k into even parts. There are as many unrestricted partitions of n-k; now sum those numbers from 0 to n to get a(n). - George Beck, Jul 22 2019

In the Young's lattice, a(n) is the number of branches that connect the (n-1)-th layer to the n-th layer. - Shouvik Datta, Sep 19 2021

a(n) is the number of multiset partitions of the multiset {r^n, s^1}, equivalently, factorization patterns of any number m=p^n*q^1 where p and q are primes. - Joerg Arndt, Jan 01 2024

a(n) is the number of positive integers whose divisors are the parts of the partitions of n + 1. - Omar E. Pol, Nov 07 2024

That is, the number of partitions of n into parts which can be listed in palindromic order.

Alternatively, number of partitions of n into parts from the set {1,2,4,6,8,10,12,...}. - T. D. Noe, Aug 05 2005

Also, partial sums of A035363.

Also number of partitions of n with at most one part occurring an odd number of times. - Reinhard Zumkeller, Dec 18 2013

The first Mathematica program computes terms of A025065; the second computes the k palindromic partitions of user-chosen n. - Clark Kimberling, Jan 20 2014

a(n) is the number of partitions p of n+1 such that 2*max(p) > n+1. - Clark Kimberling, Apr 20 2014.

From Gus Wiseman, Nov 28 2018: (Start)

Also the number of integer partitions of n + 2 that are the vertex-degrees of some hypertree. For example, the a(6) = 7 partitions of 8 that are the vertex-degrees of some hypertree, together with a realizing hypertree are:

(41111): {{1,2},{1,3},{1,4},{1,5}}

(32111): {{1,2},{1,3},{1,4},{2,5}}

(22211): {{1,2},{1,3},{2,4},{3,5}}

(311111): {{1,2},{1,3},{1,4,5,6}}

(221111): {{1,2},{1,3},{2,4,5,6}}

(2111111): {{1,2},{1,3,4,5,6,7}}

(11111111): {{1,2,3,4,5,6,7,8}}

(End)

Conjecture: a(n) is the length of maximal initial segment of A308355(n-1) that is identical to row n of A128628, for n >= 2. - Clark Kimberling, May 24 2019

From Gus Wiseman, May 21 2021: (Start)

The Heinz numbers of palindromic partitions are given by A265640. The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.

Also the number of integer partitions of n with a part greater than or equal to n/2. This is equivalent to Clark Kimberling's final comment above. The Heinz numbers of these partitions are given by A344414. For example, the a(1) = 1 through a(8) = 12 partitions are:

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

(11) (21) (22) (32) (33) (43) (44)

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

(211) (311) (51) (61) (62)

(321) (421) (71)

(411) (511) (422)

(3111) (4111) (431)

(521)

(611)

(4211)

(5111)

(41111)

Also the number of integer partitions of n with at least n/2 parts. The Heinz numbers of these partitions are given by A344296. For example, the a(1) = 1 through a(8) = 12 partitions are:

(1) (2) (21) (22) (221) (222) (2221) (2222)

(11) (111) (31) (311) (321) (3211) (3221)

(211) (2111) (411) (4111) (3311)

(1111) (11111) (2211) (22111) (4211)

(3111) (31111) (5111)

(21111) (211111) (22211)

(111111) (1111111) (32111)

(41111)

(221111)

(311111)

(2111111)

(11111111)

(End)

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)

(End)

A partition of n is a sequence p_1, ..., p_k for some k with p_1 >= p_2 >= ... >= p_k and p_1+...+p_k=n. A partition is graphical if it is the degree sequence of a simple graph (this requires that n be even). Some authors set a(0)=1 by convention.

Also number of partitions of n such that the difference between the two largest distinct parts is 2 (it is assumed that 0 is a part in each partition). Example: a(6)=3 because we have [4,2], [3,1,1,1] and [2,2,2]. - Emeric Deutsch, Apr 08 2006

Number of partitions p of n+2 such that min(p) + (number of parts of p) is a part of p. - Clark Kimberling, Feb 27 2014

Number of partitions of n+1 such that the two smallest parts differ by one. - Giovanni Resta, Mar 07 2014

Also the number of integer partitions of n with an even number of parts that cannot be grouped into pairs of distinct parts. These are also integer partitions of n with an even number of parts whose greatest multiplicity is greater than half the number of parts. - Gus Wiseman, Oct 26 2018

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

An integer partition is multigraphical if it comprises the multiset of vertex-degrees of some multigraph.

Also Heinz numbers of integer partitions of even numbers whose greatest part is less than or equal to half the sum of parts, i.e., numbers n whose sum of prime indices A056239(n) is even and at least twice the greatest prime index A061395(n). - Gus Wiseman, May 23 2021

Number of (n+1)-vertex spider graphs: trees with n+1 vertices and exactly 1 vertex of degree at least 3 (i.e. branching vertex). There is a trivial bijection with the objects described in the definition. - Emeric Deutsch, Feb 22 2014

Also the number of graphical partitions of 2n into n parts. - Gus Wiseman, Jan 08 2021

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

An integer partition is graphical if it comprises the vertex-degrees of some simple graph.

Also the number of integer partitions of n that are the vertex-degrees of some set multipartition (multiset of nonempty sets) with no singletons. - Gus Wiseman, Oct 30 2018

Naturally, such a partition must have an even number of parts. Its multiplicities form a graphical partition (A000569, A320922), and vice versa.