A338914 -id:A338914 - cp's OEIS frontend

A027187 Number of partitions of n into an even number of parts.

1, 0, 1, 1, 3, 3, 6, 7, 12, 14, 22, 27, 40, 49, 69, 86, 118, 146, 195, 242, 317, 392, 505, 623, 793, 973, 1224, 1498, 1867, 2274, 2811, 3411, 4186, 5059, 6168, 7427, 9005, 10801, 13026, 15572, 18692, 22267, 26613, 31602, 37619, 44533, 52815, 62338, 73680, 86716, 102162, 119918

Offset: 0

Clark Kimberling

nonn

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

			G.f. = 1 + x^2 + x^3 + 3*x^4 + 3*x^5 + 6*x^6 + 7*x^7 + 12*x^8 + 14*x^9 + 22*x^10 + ...
From _Gus Wiseman_, Jan 05 2021: (Start)
The a(2) = 1 through a(8) = 12 partitions into an even number of parts are the following. The Heinz numbers of these partitions are given by A028260.
  (11)  (21)  (22)    (32)    (33)      (43)      (44)
              (31)    (41)    (42)      (52)      (53)
              (1111)  (2111)  (51)      (61)      (62)
                              (2211)    (2221)    (71)
                              (3111)    (3211)    (2222)
                              (111111)  (4111)    (3221)
                                        (211111)  (3311)
                                                  (4211)
                                                  (5111)
                                                  (221111)
                                                  (311111)
                                                  (11111111)
The a(2) = 1 through a(8) = 12 partitions whose greatest part is even are the following. The Heinz numbers of these partitions are given by A244990.
  (2)  (21)  (4)    (41)    (6)      (43)      (8)
             (22)   (221)   (42)     (61)      (44)
             (211)  (2111)  (222)    (421)     (62)
                            (411)    (2221)    (422)
                            (2211)   (4111)    (431)
                            (21111)  (22111)   (611)
                                     (211111)  (2222)
                                               (4211)
                                               (22211)
                                               (41111)
                                               (221111)
                                               (2111111)
(End)

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; See p. 8, (7.323) and p. 39, Example 7.

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1000 terms from T. D. Noe)
George E. Andrews and David Newman, The Minimal Excludant in Integer Partitions, J. Int. Seq., Vol. 23 (2020), Article 20.2.3.
Arvind Ayyer, Hiranya Kishore Dey, and Digjoy Paul, How large is the character degree sum compared to the character table sum for a finite group?, arXiv:2406.06036 [math.RT], 2024. See p. 13.
Roland Bacher and Pierre De La Harpe, Conjugacy growth series of some infinitely generated groups, International Mathematics Research Notices, 2016, pp.1-53. (hal-01285685v2)
N. J. Fine, Problem 4314, Amer. Math. Monthly, Vol. 57, 1950, 421-423.
Mircea Merca, Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer, Journal of Number Theory, Volume 160, March 2016, Pages 60-75, function p_e(n).
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000701, A046682, A026838, A102462.
The Heinz numbers of these partitions are A028260.
The odd version is A027193.
The strict case is A067661.
The case of even sum as well as length is A236913 (the even bisection).
Other cases of even length:
- A024430 counts set partitions of even length.
- A034008 counts compositions of even length.
- A052841 counts ordered set partitions of even length.
- A174725 counts ordered factorizations of even length.
- A332305 counts strict compositions of even length
- A339846 counts factorizations of even length.
A000009 counts partitions into odd parts, ranked by A066208.
A026805 counts partitions whose least part is even.
A072233 counts partitions by sum and length.
A101708 counts partitions of even positive rank.
Cf. A000700, A026424, A058696, A096373, A244990, A300061.

f[n_] := Length[Select[IntegerPartitions[n], IntegerQ[First[#]/2] &]]; Table[f[n], {n, 1, 30}] (* Clark Kimberling, Mar 13 2012 *)
a[ n_] := SeriesCoefficient[ (1 + EllipticTheta[ 4, 0, x]) / (2 QPochhammer[ x]), {x, 0, n}]; (* Michael Somos, May 06 2015 *)
a[ n_] := If[ n < 0, 0, Length@Select[ IntegerPartitions[n], EvenQ[Length @ #] &]]; (* Michael Somos, May 06 2015 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( sum( k=0, sqrtint(n), (-x)^k^2, A) / eta(x + A), n))}; /* Michael Somos, Aug 19 2006 */

my(q='q+O('q^66)); Vec( (1/eta(q)+eta(q)/eta(q^2))/2 ) \\ Joerg Arndt, Mar 23 2014

a(n) = (A000041(n) + (-1)^n * A000700(n))/2.
a(n) = p(n) - p(n-1) + p(n-4) - p(n-9) + ... where p(n) is the number of unrestricted partitions of n, A000041. [Fine] - David Callan, Mar 14 2004
From Bill Gosper, Jun 25 2005: (Start)
G.f.: A(q) = Sum_{n >= 0} a(n) q^n = 1 + q^2 + q^3 + 3*q^4 + 3*q^5 + 6*q^6 + ...
  = Sum_{n >= 0} q^(2*n)/(q; q)_{2*n}
  = ((Product_{k >= 1} 1/(1-q^k)) + (Product_{k >= 1} 1/(1+q^k)))/2.
Also, let B(q) = Sum_{n >= 0} A027193(n) q^n = q + q^2 + 2*q^3 + 2*q^4 + 4*q^5 + 5*q^6 + ...
Then B(q) = Sum_{n >= 0} q^(2*n+1)/(q; q){2*n+1} = ((Product{k >= 1} 1/(1-q^k)) - (Product_{k >= 1} 1/(1+q^k)))/2.
Also we have the following identity involving 2 X 2 matrices:
Product_{k >= 1} [ 1/(1-q^(2*k)), q^k/(1-q^(2*k)) ; q^k/(1-q^(2*k)), 1/(1-q^(2*k)) ]
  = [ A(q), B(q) ; B(q), A(q) ]. (End)
a(2*n) = A046682(2*n), a(2*n+1) = A000701(2*n+1); a(n) = A000041(n)-A027193(n). - Reinhard Zumkeller, Apr 22 2006
Expansion of (1 + phi(-q)) / (2 * f(-q)) where phi(), f() are Ramanujan theta functions. - Michael Somos, Aug 19 2006
G.f.: (Sum_{k>=0} (-1)^k * x^(k^2)) / (Product_{k>0} (1 - x^k)). - Michael Somos, Aug 19 2006
a(n) = A338914(n) + A096373(n). - Gus Wiseman, Jan 06 2021

Offset changed to 0 by Michael Somos, Jul 24 2012

A025065 Number of palindromic partitions of n.

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 7, 7, 12, 12, 19, 19, 30, 30, 45, 45, 67, 67, 97, 97, 139, 139, 195, 195, 272, 272, 373, 373, 508, 508, 684, 684, 915, 915, 1212, 1212, 1597, 1597, 2087, 2087, 2714, 2714, 3506, 3506, 4508, 4508, 5763, 5763, 7338, 7338, 9296, 9296, 11732, 11732, 14742, 14742, 18460, 18460, 23025, 23025, 28629, 28629

Offset: 0

Clark Kimberling

nonn

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)

			The partitions for the first few values of n are as follows:
n: partitions .......................... number
1: 1 ................................... 1
2: 2 11 ................................ 2
3: 3 111 ............................... 2
4: 4 22 121 1111 ....................... 4
5: 5 131 212 11111 ..................... 4
6: 6 141 33 222 1221 11211 111111 ...... 7
7: 7 151 313 11311 232 21112 1111111 ... 7
From _Reinhard Zumkeller_, Jan 23 2010: (Start)
Partitions into 1,2,4,6,... for the first values of n:
1: 1 ....................................... 1
2: 2 11 .................................... 2
3: 21 111 .................................. 2
4: 4 22 211 1111 ........................... 4
5: 41 221 2111 11111 ....................... 4
6: 6 42 4211 222 2211 21111 111111.......... 7
7: 61 421 42111 2221 22111 211111 1111111 .. 7. (End)

David A. Corneth, Table of n, a(n) for n = 0..10000 (first 101 terms from Reinhard Zumkeller that were corrected by _Georg Fischer_, Jan 20 2019)

Cf. A172033, A004277. - Reinhard Zumkeller, Jan 23 2010
Cf. A004526, A030019, A056503, A147878, A320921, A322136.
The bisections are both A000070.
The ordered version (palindromic compositions) is A016116.
The complement is counted by A233771 and A210249.
The case of palindromic prime signature is A242414.
Palindromic partitions are ranked by A265640, with complement A229153.
The case of palindromic plane trees is A319436.
The multiplicative version (palindromic factorizations) is A344417.
A000569 counts graphical partitions.
A027187 counts partitions of even length, ranked by A028260.
A035363 counts partitions into even parts, ranked by A066207.
A058696 counts partitions of even numbers, ranked by A300061.
A110618 counts partitions with length <= half sum, ranked by A344291.
Cf. A000041, A067538, A143773, A209816, A338914, A338915, A340387, A344296, A344414, A344415, A344416.

a025065 = p (1:[2,4..]) where
   p [] _ = 0
   p _  0 = 1
   p ks'@(k:ks) m | m < k     = 0
                  | otherwise = p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Aug 12 2011

import Data.List (group)
a025065 = length . filter (<= 1) .
                   map (sum . map ((`mod` 2) . length) . group) . ps 1
   where ps x 0 = [[]]
         ps x y = [t:ts | t <- [x..y], ts <- ps t (y - t)]
-- Reinhard Zumkeller, Dec 18 2013

Map[Length[Select[IntegerPartitions[#], Count[OddQ[Transpose[Tally[#]][[2]]], True] <= 1 &]] &, Range[40]] (* Peter J. C. Moses, Jan 20 2014 *)
n = 8; Select[IntegerPartitions[n], Count[OddQ[Transpose[Tally[#]][[2]]], True] <= 1 &] (* Peter J. C. Moses, Jan 20 2014 *)
CoefficientList[Series[1/((1 - x) Product[1 - x^(2 n), {n, 1, 50}]), {x, 0, 60}], x] (* Clark Kimberling, Mar 14 2014 *)

N=66; q='q+O('q^N); Vec( 1/((1-q)*eta(q^2)) ) \\ Joerg Arndt, Mar 11 2014

a(n) = A000070(A004526(n)). - Reinhard Zumkeller, Jan 23 2010
G.f.: 1/((1-q)*prod(n>=1, 1-q^(2*n))). [Joerg Arndt, Mar 11 2014]
a(2*k+2) = a(2*k) + A000041(k + 1). - David A. Corneth, May 29 2021
a(n) ~ exp(Pi*sqrt(n/3)) / (2*Pi*sqrt(n)). - Vaclav Kotesovec, Nov 16 2021

Edited by N. J. A. Sloane, Dec 29 2007

Prepended a(0)=1, added more terms, Joerg Arndt, Mar 11 2014

A035363 Number of partitions of n into even parts.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 3, 0, 5, 0, 7, 0, 11, 0, 15, 0, 22, 0, 30, 0, 42, 0, 56, 0, 77, 0, 101, 0, 135, 0, 176, 0, 231, 0, 297, 0, 385, 0, 490, 0, 627, 0, 792, 0, 1002, 0, 1255, 0, 1575, 0, 1958, 0, 2436, 0, 3010, 0, 3718, 0, 4565, 0, 5604, 0, 6842, 0, 8349, 0, 10143, 0, 12310, 0

Offset: 0

Olivier Gérard

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)

			From _Joerg Arndt_, Jun 11 2013: (Start)
There are a(12)=11 symmetric unimodal compositions of 12+2=14 where the maximal part appears twice:
01:  [ 1 1 1 1 1 2 2 1 1 1 1 1 ]
02:  [ 1 1 1 1 3 3 1 1 1 1 ]
03:  [ 1 1 1 4 4 1 1 1 ]
04:  [ 1 1 2 3 3 2 1 1 ]
05:  [ 1 1 5 5 1 1 ]
06:  [ 1 2 4 4 2 1 ]
07:  [ 1 6 6 1 ]
08:  [ 2 2 3 3 2 2 ]
09:  [ 2 5 5 2 ]
10:  [ 3 4 4 3 ]
11:  [ 7 7 ]
There are a(14)=15 symmetric unimodal compositions of 14 where the maximal part appears an even number of times:
01:  [ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ]
02:  [ 1 1 1 1 1 2 2 1 1 1 1 1 ]
03:  [ 1 1 1 1 3 3 1 1 1 1 ]
04:  [ 1 1 1 2 2 2 2 1 1 1 ]
05:  [ 1 1 1 4 4 1 1 1 ]
06:  [ 1 1 2 3 3 2 1 1 ]
07:  [ 1 1 5 5 1 1 ]
08:  [ 1 2 2 2 2 2 2 1 ]
09:  [ 1 2 4 4 2 1 ]
10:  [ 1 3 3 3 3 1 ]
11:  [ 1 6 6 1 ]
12:  [ 2 2 3 3 2 2 ]
13:  [ 2 5 5 2 ]
14:  [ 3 4 4 3 ]
15:  [ 7 7 ]
(End)
a(8)=5 because we  have [8], [6,2], [4,4], [4,2,2], and [2,2,2,2]. - _Emeric Deutsch_, Jan 27 2016
From _Gus Wiseman_, May 22 2021: (Start)
The a(0) = 1 through a(12) = 11 partitions into even parts are the following (empty columns shown as dots, A = 10, C = 12). The Heinz numbers of these partitions are given by A066207.
  ()  .  (2)  .  (4)   .  (6)    .  (8)     .  (A)      .  (C)
                 (22)     (42)      (44)       (64)        (66)
                          (222)     (62)       (82)        (84)
                                    (422)      (442)       (A2)
                                    (2222)     (622)       (444)
                                               (4222)      (642)
                                               (22222)     (822)
                                                           (4422)
                                                           (6222)
                                                           (42222)
                                                           (222222)
(End)

Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem, Mathematics and Computer Education, Vol. 31, No. 1, pp. 24-28, Winter 1997. MathEduc Database (Zentralblatt MATH, 1997c.01891).
Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem II, Missouri Journal of Mathematical Sciences, Vol. 16, No. 1, Winter 2004, pp. 12-17. Zentralblatt MATH, Zbl 1071.05501.

Robert Price, Table of n, a(n) for n = 0..2001

Bisection (even part) gives the partition numbers A000041.
Column k=0 of A103919, A264398.
Cf. A036469, A000070.
Cf. A135010, A138121.
Note: A-numbers of ranking sequences are in parentheses below.
The version for odd instead of even parts is A000009 (A066208).
The version for parts divisible by 3 instead of 2 is A035377.
The strict case is A035457.
The Heinz numbers of these partitions are given by A066207.
The ordered version (compositions) is A077957 prepended by (1,0).
This is column k = 2 of A168021.
The multiplicative version (factorizations) is A340785.
A000569 counts graphical partitions (A320922).
A004526 counts partitions of length 2 (A001358).
A025065 counts palindromic partitions (A265640).
A027187 counts partitions with even length/maximum (A028260/A244990).
A058696 counts partitions of even numbers (A300061).
A067661 counts strict partitions of even length (A030229).
A236913 counts partitions of even length and sum (A340784).
A340601 counts partitions of even rank (A340602).
The following count partitions of even length:
- A096373 cannot be partitioned into strict pairs (A320891).
- A338914 can be partitioned into strict pairs (A320911).
- A338915 cannot be partitioned into distinct pairs (A320892).
- A338916 can be partitioned into distinct pairs (A320912).
- A339559 cannot be partitioned into distinct strict pairs (A320894).
- A339560 can be partitioned into distinct strict pairs (A339561).
Cf. A000041, A000290, A087897, A100484, A110618, A209816, A210249, A233771, A339004, A340385, A340387, A340786, A341447.

ZL:= [S, {C = Cycle(B), S = Set(C), E = Set(B), B = Prod(Z,Z)}, unlabelled]: seq(combstruct[count](ZL, size=n), n=0..69); # Zerinvary Lajos, Mar 26 2008
g := 1/mul(1-x^(2*k), k = 1 .. 100): gser := series(g, x = 0, 80): seq(coeff(gser, x, n), n = 0 .. 78); # Emeric Deutsch, Jan 27 2016
# Using the function EULER from Transforms (see link at the bottom of the page).
[1,op(EULER([0,1,seq(irem(n,2),n=0..66)]))]; # Peter Luschny, Aug 19 2020
# next Maple program:
a:= n-> `if`(n::odd, 0, combinat[numbpart](n/2)):
seq(a(n), n=0..84);  # Alois P. Heinz, Jun 22 2021

nmax = 50; s = Range[2, nmax, 2];
Table[Count[IntegerPartitions@n, x_ /; SubsetQ[s, x]], {n, 0, nmax}] (* Robert Price, Aug 05 2020 *)

from sympy import npartitions
def A035363(n): return 0 if n&1 else npartitions(n>>1) # Chai Wah Wu, Sep 23 2023

G.f.: Product_{k even} 1/(1 - x^k).
Convolution with the number of partitions into distinct parts (A000009, which is also number of partitions into odd parts) gives the number of partitions (A000041). - Franklin T. Adams-Watters, Jan 06 2006
If n is even then a(n)=A000041(n/2) otherwise a(n)=0. - Omar E. Pol, Nov 20 2009
G.f.: 1 + x^2*(1 - G(0))/(1-x^2) where G(k) =  1 - 1/(1-x^(2*k+2))/(1-x^2/(x^2-1/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 23 2013
a(n) = A096441(n) - A000009(n), n >= 1. - Omar E. Pol, Aug 16 2013
G.f.: exp(Sum_{k>=1} x^(2*k)/(k*(1 - x^(2*k)))). - Ilya Gutkovskiy, Aug 13 2018

A339560 Number of integer partitions of n that can be partitioned into distinct pairs of distinct parts, i.e., into a set of edges.

Original entry on oeis.org

1, 0, 0, 1, 1, 2, 2, 4, 5, 8, 8, 13, 17, 22, 28, 39, 48, 62, 81, 101, 127, 167, 202, 253, 318, 395, 486, 608, 736, 906, 1113, 1353, 1637, 2011, 2409, 2922, 3510, 4227, 5060, 6089, 7242, 8661, 10306, 12251, 14503, 17236, 20345, 24045, 28334, 33374, 39223, 46076

Offset: 0

Gus Wiseman, Dec 10 2020

nonn

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

			The a(3) = 1 through a(11) = 13 partitions (A = 10):
  (21)  (31)  (32)  (42)  (43)    (53)    (54)    (64)    (65)
              (41)  (51)  (52)    (62)    (63)    (73)    (74)
                          (61)    (71)    (72)    (82)    (83)
                          (3211)  (3221)  (81)    (91)    (92)
                                  (4211)  (3321)  (4321)  (A1)
                                          (4221)  (5221)  (4322)
                                          (4311)  (5311)  (4331)
                                          (5211)  (6211)  (4421)
                                                          (5321)
                                                          (5411)
                                                          (6221)
                                                          (6311)
                                                          (7211)
For example, the partition y = (4,3,3,2,1,1) can be partitioned into a set of edges in two ways:
  {{1,2},{1,3},{3,4}}
  {{1,3},{1,4},{2,3}},
so y is counted under a(14).

Eric Weisstein's World of Mathematics, Graphical partition.

A338916 allows equal pairs (x,x).
A339559 counts the complement in even-length partitions.
A339561 gives the Heinz numbers of these partitions.
A339619 counts factorizations of the same type.
A000070 counts non-multigraphical partitions of 2n, ranked by A339620.
A000569 counts graphical partitions, ranked by A320922.
A001358 lists semiprimes, with squarefree case A006881.
A002100 counts partitions into squarefree semiprimes.
A058696 counts partitions of even numbers, ranked by A300061.
A209816 counts multigraphical partitions, ranked by A320924.
A320655 counts factorizations into semiprimes.
A320656 counts factorizations into squarefree semiprimes.
A339617 counts non-graphical partitions of 2n, ranked by A339618.
A339655 counts non-loop-graphical partitions of 2n, ranked by A339657.
A339656 counts loop-graphical partitions, ranked by A339658.
A339659 counts graphical partitions of 2n into k parts.
The following count partitions of even length and give their Heinz numbers:
- A027187 has no additional conditions (A028260).
- A096373 cannot be partitioned into strict pairs (A320891).
- A338914 can be partitioned into strict pairs (A320911).
- A338915 cannot be partitioned into distinct pairs (A320892).
- A338916 can be partitioned into distinct pairs (A320912).
- A339559 cannot be partitioned into distinct strict pairs (A320894).
Cf. A001055, A001221, A005117, A007717, A025065, A030229, A320893, A338899, A338903, A339564.

strs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[strs[n/d],Min@@#>d&]],{d,Select[Rest[Divisors[n]],And[SquareFreeQ[#],PrimeOmega[#]==2]&]}]];
Table[Length[Select[IntegerPartitions[n],strs[Times@@Prime/@#]!={}&]],{n,0,15}]

A027187(n) = a(n) + A339559(n).

More terms from Jinyuan Wang, Feb 14 2025

A339561 Products of distinct squarefree semiprimes.

Original entry on oeis.org

1, 6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 38, 39, 46, 51, 55, 57, 58, 60, 62, 65, 69, 74, 77, 82, 84, 85, 86, 87, 90, 91, 93, 94, 95, 106, 111, 115, 118, 119, 122, 123, 126, 129, 132, 133, 134, 140, 141, 142, 143, 145, 146, 150, 155, 156, 158, 159, 161, 166

Offset: 1

Gus Wiseman, Dec 13 2020

nonn

First differs from A320911 in lacking 36.
A squarefree semiprime (A006881) is a product of any two distinct prime numbers.
The following are equivalent characteristics for any positive integer n:
(1) the prime factors of n can be partitioned into distinct strict pairs (a set of edges);
(2) n can be factored into distinct squarefree semiprimes;
(3) the prime signature of n is graphical.

			The sequence of terms together with their prime indices begins:
      1: {}        55: {3,5}         91: {4,6}
      6: {1,2}     57: {2,8}         93: {2,11}
     10: {1,3}     58: {1,10}        94: {1,15}
     14: {1,4}     60: {1,1,2,3}     95: {3,8}
     15: {2,3}     62: {1,11}       106: {1,16}
     21: {2,4}     65: {3,6}        111: {2,12}
     22: {1,5}     69: {2,9}        115: {3,9}
     26: {1,6}     74: {1,12}       118: {1,17}
     33: {2,5}     77: {4,5}        119: {4,7}
     34: {1,7}     82: {1,13}       122: {1,18}
     35: {3,4}     84: {1,1,2,4}    123: {2,13}
     38: {1,8}     85: {3,7}        126: {1,2,2,4}
     39: {2,6}     86: {1,14}       129: {2,14}
     46: {1,9}     87: {2,10}       132: {1,1,2,5}
     51: {2,7}     90: {1,2,2,3}    133: {4,8}
For example, the number 1260 can be factored into distinct squarefree semiprimes in two ways, (6*10*21) or (6*14*15), so 1260 is in the sequence. The number 69300 can be factored into distinct squarefree semiprimes in seven ways:
  (6*10*15*77)
  (6*10*21*55)
  (6*10*33*35)
  (6*14*15*55)
  (6*15*22*35)
  (10*14*15*33)
  (10*15*21*22),
so 69300 is in the sequence. A complete list of all strict factorizations of 24 is: (2*3*4), (2*12), (3*8), (4*6), (24), all of which contain at least one number that is not a squarefree semiprime, so 24 is not in the sequence.

Eric Weisstein's World of Mathematics, Graphical partition.

A309356 is a kind of universal embedding.
A320894 is the complement in A028260.
A320911 lists all (not just distinct) products of squarefree semiprimes.
A339560 counts the partitions with these Heinz numbers.
A339661 has nonzero terms at these positions.
A001358 lists semiprimes, with squarefree case A006881.
A005117 lists squarefree numbers.
A320656 counts factorizations into squarefree semiprimes.
The following count vertex-degree partitions and give their Heinz numbers:
- A058696 counts partitions of 2n (A300061).
- A000070 counts non-multigraphical partitions of 2n (A339620).
- A209816 counts multigraphical partitions (A320924).
- A320921 counts connected graphical partitions (A320923).
- A339655 counts non-loop-graphical partitions of 2n (A339657).
- A339656 counts loop-graphical partitions (A339658).
- A339617 counts non-graphical partitions of 2n (A339618).
- A000569 counts graphical partitions (A320922).
The following count partitions of even length and give their Heinz numbers:
- A027187 has no additional conditions (A028260).
- A096373 cannot be partitioned into strict pairs (A320891).
- A338914 can be partitioned into strict pairs (A320911).
- A338915 cannot be partitioned into distinct pairs (A320892).
- A338916 can be partitioned into distinct pairs (A320912).
- A339559 cannot be partitioned into distinct strict pairs (A320894).
- A339560 can be partitioned into distinct strict pairs (A339561 [this sequence]).
Cf. A001055, A001221, A002100, A007717, A030229, A112798, A320655, A320893, A338899, A338903, A339563, A339659.

sqs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sqs[n/d],Min@@#>d&]],{d,Select[Divisors[n],SquareFreeQ[#]&&PrimeOmega[#]==2&]}]];
Select[Range[100],sqs[#]!={}&]

A339561 \/ A320894 = A028260.

A339617 Number of non-graphical integer partitions of 2n.

Original entry on oeis.org

0, 1, 3, 6, 13, 25, 46, 81, 141, 234, 383, 615, 968, 1503, 2298, 3468, 5176, 7653, 11178, 16212, 23290, 33218, 46996, 66091, 92277, 128122, 176787, 242674, 331338, 450279, 608832, 819748, 1098907, 1467122, 1951020, 2584796, 3411998

Offset: 0

Gus Wiseman, Dec 13 2020

nonn

An integer partition is graphical if it comprises the multiset of vertex-degrees of some graph. See A209816 for multigraphical partitions, A000070 for non-multigraphical partitions. Graphical partitions are counted by A000569.
The following are equivalent characteristics for any positive integer n:
(1) the prime indices of n can be partitioned into distinct strict pairs (a set of edges);
(2) n can be factored into distinct squarefree semiprimes;
(3) the prime signature of n is graphical.

			The a(1) = 1 through a(4) = 13 partitions:
  (2)  (4)    (6)      (8)
       (2,2)  (3,3)    (4,4)
       (3,1)  (4,2)    (5,3)
              (5,1)    (6,2)
              (3,2,1)  (7,1)
              (4,1,1)  (3,3,2)
                       (4,2,2)
                       (4,3,1)
                       (5,2,1)
                       (6,1,1)
                       (3,3,1,1)
                       (4,2,1,1)
                       (5,1,1,1)
For example, the partition (2,2,2,2) is not counted under a(4) because there are three possible graphs with the prescribed degrees:
  {{1,2},{1,3},{2,4},{3,4}}
  {{1,2},{1,4},{2,3},{3,4}}
  {{1,3},{1,4},{2,3},{2,4}}

Eric Weisstein's World of Mathematics, Graphical partition.

A006881 lists squarefree semiprimes.
A320656 counts factorizations into squarefree semiprimes.
A339659 counts graphical partitions of 2n into k parts.
The following count vertex-degree partitions and give their Heinz numbers:
- A058696 counts partitions of 2n (A300061).
- A000070 counts non-multigraphical partitions of 2n (A339620).
- A209816 counts multigraphical partitions (A320924).
- A339655 counts non-loop-graphical partitions of 2n (A339657).
- A339656 counts loop-graphical partitions (A339658).
- A339617 [this sequence] counts non-graphical partitions of 2n (A339618).
- A000569 counts graphical partitions (A320922).
The following count partitions of even length and give their Heinz numbers:
- A027187 has no additional conditions (A028260).
- A096373 cannot be partitioned into strict pairs (A320891).
- A338914 can be partitioned into strict pairs (A320911).
- A338915 cannot be partitioned into distinct pairs (A320892).
- A338916 can be partitioned into distinct pairs (A320912).
- A339559 cannot be partitioned into distinct strict pairs (A320894).
- A339560 can be partitioned into distinct strict pairs (A339561).
Cf. A001055, A007717, A025065, A320921, A320922, A338899, A339564, A339619, A339660, A339661.

prptns[m_]:=Union[Sort/@If[Length[m]==0,{{}},Join@@Table[Prepend[#,m[[ipr]]]&/@prptns[Delete[m,List/@ipr]],{ipr,Select[Prepend[{#},1]&/@Select[Range[2,Length[m]],m[[#]]>m[[#-1]]&],UnsameQ@@m[[#]]&]}]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
Table[Length[Select[strnorm[2*n],Select[prptns[#],UnsameQ@@#&]=={}&]],{n,0,5}]

a(n) + A000569(n) = A000041(2*n).

A338915 Number of integer partitions of n that have an even number of parts and cannot be partitioned into distinct pairs of not necessarily distinct parts.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 1, 4, 2, 6, 6, 12, 12, 20, 22, 38, 42, 60, 73, 101, 124, 164, 203, 266, 319, 415, 507, 649, 786, 983, 1198, 1499, 1797, 2234, 2673, 3303, 3952, 4826, 5753, 6999, 8330, 10051, 11943, 14357, 16956, 20322, 23997, 28568, 33657, 39897, 46879

Offset: 0

Gus Wiseman, Dec 10 2020

nonn

The multiplicities of such a partition form a non-loop-graphical partition (A339655, A339657).

			The a(7) = 1 through a(12) = 12 partitions:
  211111  2222      411111    222211      222221      3333
          221111    21111111  331111      611111      222222
          311111              511111      22211111    441111
          11111111            22111111    32111111    711111
                              31111111    41111111    22221111
                              1111111111  2111111111  32211111
                                                      33111111
                                                      42111111
                                                      51111111
                                                      2211111111
                                                      3111111111
                                                      111111111111
For example, the partition y = (3,2,2,1,1,1,1,1) can be partitioned into pairs in just three ways:
  {{1,1},{1,1},{1,2},{2,3}}
  {{1,1},{1,1},{1,3},{2,2}}
  {{1,1},{1,2},{1,2},{1,3}}
None of these is strict, so y is counted under a(12).

Eric Weisstein's World of Mathematics, Graphical partition.

The Heinz numbers of these partitions are A320892.
The complement in even-length partitions is A338916.
A000070 counts non-multigraphical partitions of 2n, ranked by A339620.
A000569 counts graphical partitions, ranked by A320922.
A001358 lists semiprimes, with squarefree case A006881.
A058696 counts partitions of even numbers, ranked by A300061.
A209816 counts multigraphical partitions, ranked by A320924.
A320655 counts factorizations into semiprimes.
A322353 counts factorizations into distinct semiprimes.
A339617 counts non-graphical partitions of 2n, ranked by A339618.
A339655 counts non-loop-graphical partitions of 2n, ranked by A339657.
A339656 counts loop-graphical partitions, ranked by A339658.
The following count partitions of even length and give their Heinz numbers:
- A027187 has no additional conditions (A028260).
- A096373 cannot be partitioned into strict pairs (A320891).
- A338914 can be partitioned into strict pairs (A320911).
- A338916 can be partitioned into distinct pairs (A320912).
- A339559 cannot be partitioned into distinct strict pairs (A320894).
- A339560 can be partitioned into distinct strict pairs (A339561).
Cf. A001055, A007717, A025065, A320656, A320732, A320893, A338898, A338902.

smcs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[smcs[n/d],Min@@#>d&]],{d,Select[Rest[Divisors[n]],PrimeOmega[#]==2&]}]];
Table[Length[Select[IntegerPartitions[n],EvenQ[Length[#]]&&smcs[Times@@Prime/@#]=={}&]],{n,0,10}]

A027187(n) = a(n) + A338916(n).

More terms from Jinyuan Wang, Feb 14 2025

A339618 Heinz numbers of non-graphical integer partitions of even numbers.

Original entry on oeis.org

3, 7, 9, 10, 13, 19, 21, 22, 25, 28, 29, 30, 34, 37, 39, 43, 46, 49, 52, 53, 55, 57, 61, 62, 63, 66, 70, 71, 75, 76, 79, 82, 84, 85, 87, 88, 89, 91, 94, 100, 101, 102, 107, 111, 113, 115, 116, 117, 118, 121, 129, 130, 131, 133, 134, 136, 138, 139, 146, 147

Offset: 1

Gus Wiseman, Dec 18 2020

nonn

An integer partition is graphical if it comprises the multiset of vertex-degrees of some graph. Graphical partitions are counted by A000569.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.
The following are equivalent characteristics for any positive integer n:
(1) the multiset of prime indices of n can be partitioned into distinct strict pairs (a set of edges);
(2) n can be factored into distinct squarefree semiprimes;
(3) the unordered prime signature of n is graphical.

			The sequence of terms together with their prime indices begins:
      3: {2}         43: {14}        79: {22}
      7: {4}         46: {1,9}       82: {1,13}
      9: {2,2}       49: {4,4}       84: {1,1,2,4}
     10: {1,3}       52: {1,1,6}     85: {3,7}
     13: {6}         53: {16}        87: {2,10}
     19: {8}         55: {3,5}       88: {1,1,1,5}
     21: {2,4}       57: {2,8}       89: {24}
     22: {1,5}       61: {18}        91: {4,6}
     25: {3,3}       62: {1,11}      94: {1,15}
     28: {1,1,4}     63: {2,2,4}    100: {1,1,3,3}
     29: {10}        66: {1,2,5}    101: {26}
     30: {1,2,3}     70: {1,3,4}    102: {1,2,7}
     34: {1,7}       71: {20}       107: {28}
     37: {12}        75: {2,3,3}    111: {2,12}
     39: {2,6}       76: {1,1,8}    113: {30}
For example, there are three possible multigraphs with degrees (1,1,3,3):
  {{1,2},{1,2},{1,2},{3,4}}
  {{1,2},{1,2},{1,3},{2,4}}
  {{1,2},{1,2},{1,4},{2,3}}.
Since none of these is a graph, the Heinz number 100 belongs to the sequence.

Eric Weisstein's World of Mathematics, Graphical partition.

A181819 applied to A320894 gives this sequence.
A300061 is a superset.
A339617 counts these partitions.
A320922 ranks the complement, counted by A000569.
A006881 lists squarefree semiprimes.
A320656 counts factorizations into squarefree semiprimes.
A339659 counts graphical partitions of 2n into k parts.
The following count vertex-degree partitions and give their Heinz numbers:
- A058696 counts partitions of 2n (A300061).
- A000070 counts non-multigraphical partitions of 2n (A339620).
- A209816 counts multigraphical partitions (A320924).
- A339655 counts non-loop-graphical partitions of 2n (A339657).
- A339656 counts loop-graphical partitions (A339658).
- A339617 counts non-graphical partitions of 2n (A339618 [this sequence]).
- A000569 counts graphical partitions (A320922).
The following count partitions of even length and give their Heinz numbers:
- A027187 has no additional conditions (A028260).
- A096373 cannot be partitioned into strict pairs (A320891).
- A338914 can be partitioned into strict pairs (A320911).
- A338915 cannot be partitioned into distinct pairs (A320892).
- A338916 can be partitioned into distinct pairs (A320912).
- A339559 cannot be partitioned into distinct strict pairs (A320894).
- A339560 can be partitioned into distinct strict pairs (A339561).
Cf. A001358, A007717, A050326, A320923, A338899.

strs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[strs[n/d],Min@@#>d&]],{d,Select[Divisors[n],And[SquareFreeQ[#],PrimeOmega[#]==2]&]}]];
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],EvenQ[Length[nrmptn[#]]]&&strs[Times@@Prime/@nrmptn[#]]=={}&]

Equals A300061 \ A320922.

For all n, A181821(a(n)) and A304660(a(n)) belong to A320894.

A338916 Number of integer partitions of n that can be partitioned into distinct pairs of (possibly equal) parts.

Original entry on oeis.org

1, 0, 1, 1, 2, 3, 5, 6, 8, 12, 16, 21, 28, 37, 49, 64, 80, 104, 135, 169, 216, 268, 341, 420, 527, 654, 809, 991, 1218, 1488, 1828, 2213, 2687, 3262, 3934, 4754, 5702, 6849, 8200, 9819, 11693, 13937, 16562, 19659, 23262, 27577, 32493, 38341, 45112, 53059, 62265

Offset: 0

Gus Wiseman, Dec 10 2020

nonn

The multiplicities of such a partition form a loop-graphical partition (A339656, A339658).

			The a(2) = 1 through a(10) = 16 partitions:
  (11)  (21)  (22)  (32)    (33)    (43)    (44)    (54)      (55)
              (31)  (41)    (42)    (52)    (53)    (63)      (64)
                    (2111)  (51)    (61)    (62)    (72)      (73)
                            (2211)  (2221)  (71)    (81)      (82)
                            (3111)  (3211)  (3221)  (3222)    (91)
                                    (4111)  (3311)  (3321)    (3322)
                                            (4211)  (4221)    (3331)
                                            (5111)  (4311)    (4222)
                                                    (5211)    (4321)
                                                    (6111)    (4411)
                                                    (222111)  (5221)
                                                    (321111)  (5311)
                                                              (6211)
                                                              (7111)
                                                              (322111)
                                                              (421111)
For example, the partition (4,2,1,1,1,1) can be partitioned into {{1,1},{1,2},{1,4}}, and thus is counted under a(10).

Eric Weisstein's World of Mathematics, Graphical partition.

A320912 gives the Heinz numbers of these partitions.
A338915 counts the complement in even-length partitions.
A339563 counts factorizations of the same type.
A000070 counts non-multigraphical partitions of 2n, ranked by A339620.
A000569 counts graphical partitions, ranked by A320922.
A001358 lists semiprimes, with squarefree case A006881.
A058696 counts partitions of even numbers, ranked by A300061.
A209816 counts multigraphical partitions, ranked by A320924.
A320655 counts factorizations into semiprimes.
A322353 counts factorizations into distinct semiprimes.
A339617 counts non-graphical partitions of 2n, ranked by A339618.
A339655 counts non-loop-graphical partitions of 2n, ranked by A339657.
A339656 counts loop-graphical partitions, ranked by A339658.
The following count partitions of even length and give their Heinz numbers:
- A027187 has no additional conditions (A028260).
- A096373 cannot be partitioned into strict pairs (A320891).
- A338914 can be partitioned into strict pairs (A320911).
- A338915 cannot be partitioned into distinct pairs (A320892).
- A339559 cannot be partitioned into distinct strict pairs (A320894).
- A339560 can be partitioned into distinct strict pairs (A339561).
Cf. A001055, A007717, A025065, A320656, A320732, A320893, A320921, A338898, A338902, A339564.

stfs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[stfs[n/d],Min@@#>d&]],{d,Select[Rest[Divisors[n]],PrimeOmega[#]==2&]}]];
Table[Length[Select[IntegerPartitions[n],stfs[Times@@Prime/@#]!={}&]],{n,0,20}]

A027187(n) = a(n) + A338915(n).

More terms from Jinyuan Wang, Feb 14 2025

A339655 Number of non-loop-graphical integer partitions of 2n.

Original entry on oeis.org

0, 0, 1, 3, 7, 14, 28, 51, 91, 156, 260, 425, 680, 1068, 1654, 2524, 3802, 5668, 8350, 12190, 17634, 25306, 36011, 50902, 71441, 99642

Offset: 0

Gus Wiseman, Dec 14 2020

An integer partition is loop-graphical if it comprises the multiset of vertex-degrees of some graph with loops, where a loop is an edge with equal source and target. See A339657 for the Heinz numbers, and A339656 for the complement.
The following are equivalent characteristics for any positive integer n:
(1) the prime factors of n can be partitioned into distinct pairs;
(2) n can be factored into distinct semiprimes;
(3) the prime signature of n is loop-graphical.

			The a(2) = 1 through a(5) = 14 partitions (A = 10):
  (4)  (6)    (8)      (A)
       (4,2)  (4,4)    (5,5)
       (5,1)  (5,3)    (6,4)
              (6,2)    (7,3)
              (7,1)    (8,2)
              (5,2,1)  (9,1)
              (6,1,1)  (5,3,2)
                       (5,4,1)
                       (6,2,2)
                       (6,3,1)
                       (7,2,1)
                       (8,1,1)
                       (6,2,1,1)
                       (7,1,1,1)
For example, the seven normal loop-multigraphs with degrees y = (5,3,2) are:
  {{1,1},{1,1},{1,2},{2,2},{3,3}}
  {{1,1},{1,1},{1,2},{2,3},{2,3}}
  {{1,1},{1,1},{1,3},{2,2},{2,3}}
  {{1,1},{1,2},{1,2},{1,2},{3,3}}
  {{1,1},{1,2},{1,2},{1,3},{2,3}}
  {{1,1},{1,2},{1,3},{1,3},{2,2}}
  {{1,2},{1,2},{1,2},{1,3},{1,3}},
but since none of these is a loop-graph (because they are not strict), y is counted under a(5).

Eric Weisstein's World of Mathematics, Graphical partition.

A001358 lists semiprimes, with squarefree case A006881.
A006125 counts labeled graphs, with covering case A006129.
A062740 counts labeled connected loop-graphs.
A101048 counts partitions into semiprimes.
A320461 ranks normal loop-graphs.
A322661 counts covering loop-graphs.
A320655 counts factorizations into semiprimes.
The following count vertex-degree partitions and give their Heinz numbers:
- A058696 counts partitions of 2n (A300061).
- A000070 counts non-multigraphical partitions of 2n (A339620).
- A209816 counts multigraphical partitions (A320924).
- A339655 (this sequence) counts non-loop-graphical partitions of 2n (A339657).
- A339656 counts loop-graphical partitions (A339658).
- A339617 counts non-graphical partitions of 2n (A339618).
- A000569 counts graphical partitions (A320922).
The following count partitions of even length and give their Heinz numbers:
- A027187 has no additional conditions (A028260).
- A096373 cannot be partitioned into strict pairs (A320891).
- A338914 can be partitioned into strict pairs (A320911).
- A338915 cannot be partitioned into distinct pairs (A320892).
- A338916 can be partitioned into distinct pairs (A320912).
- A339559 cannot be partitioned into distinct strict pairs (A320894).
- A339560 can be partitioned into distinct strict pairs (A339561).
Cf. A007717, A025065, A147878, A320732, A320921, A338898, A338902, A338912, A338913, A339659, A339660, A339662.

spsbin[{}]:={{}};spsbin[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@spsbin[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mpsbin[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@spsbin[Range[Length[set]]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
Table[Length[Select[strnorm[2*n],Select[mpsbin[#],UnsameQ@@#&]=={}&]],{n,0,5}]

A058696(n) = a(n) + A339656(n).

a(7)-a(25) from Andrew Howroyd, Jan 10 2024

cp's OEIS Frontend

A027187 Number of partitions of n into an even number of parts.

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A025065 Number of palindromic partitions of n.

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A035363 Number of partitions of n into even parts.

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A339560 Number of integer partitions of n that can be partitioned into distinct pairs of distinct parts, i.e., into a set of edges.

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A339561 Products of distinct squarefree semiprimes.

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A339617 Number of non-graphical integer partitions of 2n.

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A338915 Number of integer partitions of n that have an even number of parts and cannot be partitioned into distinct pairs of not necessarily distinct parts.