A236559 -id:A236559 - cp's OEIS frontend

A347449 Number of integer partitions of n with reverse-alternating product > 1.

0, 0, 1, 1, 2, 2, 5, 5, 10, 11, 20, 22, 37, 41, 66, 75, 113, 129, 190, 218, 310, 358, 497, 576, 782, 908, 1212, 1411, 1851, 2156, 2793, 3255, 4163, 4853, 6142, 7159, 8972, 10451, 12989, 15123, 18646, 21689, 26561, 30867, 37556, 43599, 52743, 61161, 73593

Offset: 0

Gus Wiseman, Sep 16 2021

nonn

All such partitions have odd length.

We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)). The reverse-alternating product is the alternating product of the reversed sequence.

			The a(2) = 1 through a(9) = 11 partitions:
  (2)  (3)  (4)    (5)    (6)      (7)      (8)        (9)
            (211)  (311)  (222)    (322)    (332)      (333)
                          (321)    (421)    (422)      (432)
                          (411)    (511)    (431)      (522)
                          (21111)  (31111)  (521)      (531)
                                            (611)      (621)
                                            (22211)    (711)
                                            (32111)    (32211)
                                            (41111)    (42111)
                                            (2111111)  (51111)
                                                       (3111111)

The strict case is A067659, except that a(0) = a(1) = 0.
The even bisection is A236559.
The non-reverse multiplicative version is A339890, weak A347456.
The case of >= 1 instead of > 1 is A344607.
The opposite version is A344608, also the non-reverse even-length case.
The complement is counted by A347443, non-reverse A119620.
Allowing any integer reverse-alternating product gives A347445.
Allowing any integer alternating product gives A347446.
Reverse version of A347448; also the odd-length case.
The Heinz numbers of these partitions are the complement of A347450.
The multiplicative version (factorizations) is A347705.
A000041 counts partitions.
A027187 counts partitions of even length.
A027193 counts partitions of odd length.
A100824 counts partitions of n with alternating sum <= 1.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A347462 counts possible reverse-alternating products of partitions.
Cf. A000070, A008549, A086543, A182616, A236913, A325534, A325535, A344611, A347442, A347444, A347447, A347453, A347461, A347465.

altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
Table[Length[Select[IntegerPartitions[n],altprod[Reverse[#]]>1&]],{n,0,30}]

a(n) = A344607(n) - A119620(n).

A352129 Number of strict integer partitions of n with as many even conjugate parts as odd conjugate parts.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 2, 1, 3, 2, 3, 4, 3, 5, 5, 6, 6, 9, 8, 10, 12, 13, 15, 17, 20, 20, 26, 26, 32, 35, 39, 44, 50, 55, 61, 71, 76, 87, 96, 108, 117, 135, 145, 164, 181, 200, 222, 246, 272, 298, 334, 363, 404, 443

Offset: 0

Gus Wiseman, Mar 15 2022

nonn

			The a(n) strict partitions for selected n:
n = 3      13         15         18         20           22
   ------------------------------------------------------------------
    (2,1)  (6,5,2)    (10,5)     (12,6)     (12,7,1)     (12,8,2)
           (6,4,2,1)  (6,4,3,2)  (8,7,3)    (8,5,4,3)    (8,6,5,3)
                      (6,5,3,1)  (8,5,3,2)  (8,6,4,2)    (8,7,5,2)
                                 (8,6,3,1)  (8,7,4,1)    (12,7,2,1)
                                            (8,6,3,2,1)  (8,6,4,3,1)
                                                         (8,7,4,2,1)

This is the strict case of A045931, ranked by A350848 (zeros of A350941).
The conjugate version is A239241, non-strict A045931 (ranked by A325698).
A000041 counts integer partitions, strict A000009.
A130780 counts partitions with no more even than odd parts, strict A239243.
A171966 counts partitions with no more odd than even parts, strict A239240.
There are four statistics:
- A257991 = # of odd parts, conjugate A344616.
- A257992 = # of even parts, conjugate A350847.
There are four other pairings of statistics:
- A277579, ranked by A349157, strict A352131.
- A277103, ranked by A350944.
- A277579, ranked by A350943, strict A352130.
- A350948, ranked by A350945.
There are three double-pairings of statistics:
- A351976, ranked by A350949.
- A351977, ranked by A350946, strict A352128.
- A351981, ranked by A351980.
The case of all four statistics equal is A351978, ranked by A350947.
Cf. A000070, A098123, A195017, A236559, A241638, A350839, A350849, A350942.

conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Count[conj[#],?OddQ]==Count[conj[#],?EvenQ]&]],{n,0,30}]

A100824 Number of partitions of n with at most one odd part.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 3, 7, 5, 12, 7, 19, 11, 30, 15, 45, 22, 67, 30, 97, 42, 139, 56, 195, 77, 272, 101, 373, 135, 508, 176, 684, 231, 915, 297, 1212, 385, 1597, 490, 2087, 627, 2714, 792, 3506, 1002, 4508, 1255, 5763, 1575, 7338, 1958, 9296, 2436, 11732, 3010, 14742

Offset: 0

Vladeta Jovovic, Jan 13 2005

From Gus Wiseman, Jan 21 2022: (Start)
Also the number of integer partitions of n with alternating sum <= 1, where the alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i. These are the conjugates of partitions with at most one odd part. For example, the a(1) = 1 through a(9) = 12 partitions with alternating sum <= 1 are:
  1  11  21   22    32     33      43       44        54
         111  1111  221    2211    331      2222      441
                    2111   111111  2221     3311      3222
                    11111          3211     221111    3321
                                   22111    11111111  4311
                                   211111             22221
                                   1111111            33111
                                                      222111
                                                      321111
                                                      2211111
                                                      21111111
                                                      111111111
(End)

			From _Gus Wiseman_, Jan 21 2022: (Start)
The a(1) = 1 through a(9) = 12 partitions with at most one odd part:
  (1)  (2)  (3)   (4)   (5)    (6)    (7)     (8)     (9)
            (21)  (22)  (32)   (42)   (43)    (44)    (54)
                        (41)   (222)  (52)    (62)    (63)
                        (221)         (61)    (422)   (72)
                                      (322)   (2222)  (81)
                                      (421)           (432)
                                      (2221)          (441)
                                                      (522)
                                                      (621)
                                                      (3222)
                                                      (4221)
                                                      (22221)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A008951, A000070, A000097, A000098, A000710.
The case of alternating sum 0 (equality) is A000070.
A multiplicative version is A339846.
These partitions are ranked by A349150, conjugate A349151.
A000041 = integer partitions, strict A000009.
A027187 = partitions of even length, strict A067661, ranked by A028260.
A027193 = partitions of odd length, ranked by A026424.
A058695 = partitions of odd numbers.
A103919 = partitions by sum and alternating sum (reverse: A344612).
A277103 = partitions with the same number of odd parts as their conjugate.
Cf. A000984, A001791, A008549, A097805, A119620, A182616, A236559, A236913, A236914, A304620, A344607, A345958, A347443.

seq(coeff(convert(series((1+x/(1-x^2))/mul(1-x^(2*i),i=1..100),x,100),polynom),x,n),n=0..60); (C. Ronaldo)

nmax = 50; CoefficientList[Series[(1+x/(1-x^2)) * Product[1/(1-x^(2*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 07 2016 *)
Table[Length[Select[IntegerPartitions[n],Count[#,?OddQ]<=1&]],{n,0,30}] (* _Gus Wiseman, Jan 21 2022 *)

a(n) = if(n%2==0, numbpart(n/2), sum(i=1, (n+1)\2, numbpart((n-2*i+1)\2))) \\ David A. Corneth, Jan 23 2022

G.f.: (1+x/(1-x^2))/Product(1-x^(2*i), i=1..infinity). More generally, g.f. for number of partitions of n with at most k odd parts is (1+Sum(x^i/Product(1-x^(2*j), j=1..i), i=1..k))/Product(1-x^(2*i), i=1..infinity).
a(n) ~ exp(sqrt(n/3)*Pi) / (2*sqrt(3)*n) if n is even and a(n) ~ exp(sqrt(n/3)*Pi) / (2*Pi*sqrt(n)) if n is odd. - Vaclav Kotesovec, Mar 07 2016
a(2*n) = A000041(n). a(2*n + 1) = A000070(n). - David A. Corneth, Jan 23 2022

More terms from C. Ronaldo (aga_new_ac(AT)hotmail.com), Jan 19 2005

A352130 Number of strict integer partitions of n with as many odd parts as even conjugate parts.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 4, 5, 6, 7, 7, 8, 9, 11, 12, 13, 14, 16, 18, 21, 23, 25, 28, 31, 34, 37, 41, 45, 50, 55, 60, 65, 72, 79, 86, 93, 102, 111, 121, 132, 143, 155, 169, 183, 197, 213, 231, 251, 271, 292, 315, 340, 367, 396

Offset: 0

Gus Wiseman, Mar 15 2022

nonn

			The a(n) strict partitions for selected n:
n = 2    7        9        13        14         15         16
   --------------------------------------------------------------------
    (2)  (6,1)    (8,1)    (12,1)    (14)       (14,1)     (16)
         (4,2,1)  (4,3,2)  (6,4,3)   (6,5,3)    (6,5,4)    (8,5,3)
                  (6,2,1)  (8,3,2)   (10,3,1)   (8,4,3)    (12,3,1)
                           (10,2,1)  (6,4,3,1)  (10,3,2)   (6,5,4,1)
                                     (8,3,2,1)  (12,2,1)   (8,4,3,1)
                                                (6,5,3,1)  (10,3,2,1)
                                                           (6,4,3,2,1)

This is the strict case of A277579, ranked by A350943 (zeros of A350942).
The conjugate version is A352131, non-strict A277579 (ranked by A349157).
A000041 counts integer partitions, strict A000009.
A130780 counts partitions with no more even than odd parts, strict A239243.
A171966 counts partitions with no more odd than even parts, strict A239240.
There are four statistics:
- A257991 = # of odd parts, conjugate A344616.
- A257992 = # of even parts, conjugate A350847.
There are four other pairings of statistics:
- A045931, ranked by A325698, strict A239241.
- A045931, ranked by A350848, strict A352129.
- A277103, ranked by A350944, strict new.
- A350948, ranked by A350945, strict new.
There are three double-pairings of statistics:
- A351976, ranked by A350949, strict A010054?
- A351977, ranked by A350946, strict A352128.
- A351981, ranked by A351980. strict A014105?
The case of all four statistics equal is A351978, ranked by A350947.
Cf. A027187, A027193, A103919, A122111, A236559, A325039, A344607, A344651, A345196, A350950, A350951.

conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Count[#,?OddQ]==Count[conj[#],?EvenQ]&]],{n,0,30}]

A304620 Expansion of (1/(1 - x)) * Sum_{k>=0} x^(2k) / Product_{j=1..2k} (1 - x^j).

Original entry on oeis.org

1, 1, 2, 3, 6, 9, 15, 22, 34, 48, 70, 97, 137, 186, 255, 341, 459, 605, 800, 1042, 1359, 1751, 2256, 2879, 3672, 4645, 5869, 7367, 9234, 11508, 14319, 17730, 21916, 26975, 33143, 40570, 49575, 60376, 73402, 88974, 107666, 129933, 156546, 188148, 225767, 270300, 323115, 385453

Offset: 0

Ilya Gutkovskiy, Aug 19 2018

nonn

Partial sums of A027187.
From Gus Wiseman, Jun 26 2021: (Start)
Also the number of integer partitions of 2n+1 with odd greatest part and alternating sum 1, where the alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i, which is equal to the number of odd parts in the conjugate partition. For example, the a(0) = 1 through a(6) = 15 partitions are:
  1   111   32      331       54          551           76
            11111   3211      3222        3332          5422
                    1111111   3321        5411          5521
                              33111       33221         33331
                              321111      322211        55111
                              111111111   332111        322222
                                          3311111       332221
                                          32111111      333211
                                          11111111111   541111
                                                        3322111
                                                        32221111
                                                        33211111
                                                        331111111
                                                        3211111111
                                                        1111111111111
Also odd-length partitions of 2n+1 with exactly one odd part.
(End)

Joerg Arndt, Matters Computational (The Fxtbook), section 16.4.1 "Unrestricted partitions and partitions into m parts", page 347.
Index entries for sequences related to partitions

First differences are A027187.
The version for even instead of odd greatest part is A306145.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A000070 counts partitions with alternating sum 1.
A067661 counts strict partitions of even length.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A344610 counts partitions by sum and positive reverse-alternating sum.
Cf. A000097, A006330, A027193, A030229, A067659, A236559, A236914, A239829, A239830, A318156, A338907, A344611.

nmax = 47; CoefficientList[Series[1/(1 - x) Sum[x^(2 k)/Product[(1 - x^j), {j, 1, 2 k}], {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 47; CoefficientList[Series[(1 + EllipticTheta[4, 0, x])/(2 (1 - x) QPochhammer[x]), {x, 0, nmax}], x]
Table[Length[Select[IntegerPartitions[n],OddQ[Length[#]]&&Count[#,?OddQ]==1&]],{n,1,30,2}] (* _Gus Wiseman, Jun 26 2021 *)

a(n) = A000070(n) - A306145(n).

a(n) ~ exp(Pi*sqrt(2*n/3)) / (2^(5/2)*Pi*sqrt(n)). - Vaclav Kotesovec, Aug 20 2018

A352131 Number of strict integer partitions of n with same number of even parts as odd conjugate parts.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 2, 3, 2, 2, 3, 4, 3, 4, 5, 5, 5, 6, 7, 7, 8, 10, 10, 10, 12, 14, 15, 14, 17, 21, 20, 20, 25, 28, 28, 29, 34, 39, 39, 40, 47, 52, 53, 56, 64, 70, 71, 77, 86, 92, 97, 104, 114, 122

Offset: 0

Gus Wiseman, Mar 15 2022

nonn

			The a(n) strict partitions for selected n:
n = 3      10         14         18         21             24
   ----------------------------------------------------------------------
    (2,1)  (6,4)      (8,6)      (10,8)     (11,10)        (8,7,5,4)
           (4,3,2,1)  (5,4,3,2)  (6,5,4,3)  (8,6,4,3)      (9,8,4,3)
                      (6,5,2,1)  (7,6,3,2)  (8,7,4,2)      (10,8,4,2)
                                 (8,7,2,1)  (10,8,2,1)     (10,9,3,2)
                                            (6,5,4,3,2,1)  (11,10,2,1)
                                                           (8,6,4,3,2,1)

This is the strict case of A277579, ranked by A349157 (zeros of A350849).
The conjugate version is A352130, non-strict A277579 (ranked by A350943).
A000041 counts integer partitions, strict A000009.
A130780 counts partitions with no more even than odd parts, strict A239243.
A171966 counts partitions with no more odd than even parts, strict A239240.
There are four statistics:
- A257991 = # of odd parts, conjugate A344616.
- A257992 = # of even parts, conjugate A350847.
There are four other pairings of statistics:
- A045931, ranked by A325698, strict A239241.
- A045931, ranked by A350848, strict A352129.
- A277103, ranked by A350944.
- A350948, ranked by A350945.
There are three double-pairings of statistics:
- A351976, ranked by A350949.
- A351977, ranked by A350946, strict A352128.
- A351981, ranked by A351980.
The case of all four statistics equal is A351978, ranked by A350947.
Cf. A027187, A027193, A103919, A122111, A236559, A325039, A344607, A344651, A345196, A350942, A350950, A350951.

conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Count[#,?EvenQ]==Count[conj[#],?OddQ]&]],{n,0,30}]

A306145 Expansion of (1/(1 - x)) * Sum_{k>=0} x^(2k+1) / Product_{j=1..2k+1} (1 - x^j).

Original entry on oeis.org

0, 1, 2, 4, 6, 10, 15, 23, 33, 49, 69, 98, 135, 187, 253, 343, 456, 607, 797, 1045, 1355, 1755, 2252, 2884, 3666, 4651, 5863, 7375, 9226, 11517, 14310, 17741, 21904, 26988, 33130, 40586, 49558, 60394, 73383, 88996, 107642, 129958, 156519, 188178, 225734, 270335, 323078, 385494

Offset: 0

Ilya Gutkovskiy, Aug 19 2018

nonn

Partial sums of A027193.
From Gus Wiseman, Jun 23 2021: (Start)
Also the number of even-length integer partitions of 2n+1 with exactly one odd part. For example, the a(1) = 1 through a(5) = 10 partitions are:
  (2,1)  (3,2)  (4,3)      (5,4)      (6,5)
         (4,1)  (5,2)      (6,3)      (7,4)
                (6,1)      (7,2)      (8,3)
                (2,2,2,1)  (8,1)      (9,2)
                           (3,2,2,2)  (10,1)
                           (4,2,2,1)  (4,3,2,2)
                                      (4,4,2,1)
                                      (5,2,2,2)
                                      (6,2,2,1)
                                      (2,2,2,2,2,1)
Also partitions of 2n+1 with even greatest part and alternating sum 1.
(End)

Joerg Arndt, Matters Computational (The Fxtbook), section 16.4.1 "Unrestricted partitions and partitions into m parts", page 347.
Index entries for sequences related to partitions

First differences are A027193.
The ordered version appears to be A087447 modulo initial terms.
The version for odd instead of even-length partitions is A304620.
The case of strict partitions is A318156.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A027187 counts partitions of even length, with strict case A067661.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A344610 counts partitions by sum and positive reverse-alternating sum.
Cf. A000070, A000097, A006330, A030229, A067659, A236559, A236914, A239829, A239830, A338907, A344611.

nmax = 47; CoefficientList[Series[1/(1 - x) Sum[x^(2 k + 1)/Product[(1 - x^j), {j, 1, 2 k + 1}], {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 47; CoefficientList[Series[(1 - EllipticTheta[4, 0, x])/(2 (1 - x) QPochhammer[x]), {x, 0, nmax}], x]
Table[Length[Select[IntegerPartitions[n],EvenQ[Length[#]]&&Count[#,?OddQ]==1&]],{n,1,30,2}] (* _Gus Wiseman, Jun 23 2021 *)

a(n) = A000070(n) - A304620(n).

a(n) ~ exp(Pi*sqrt(2*n/3)) / (2^(5/2)*Pi*sqrt(n)). - Vaclav Kotesovec, Aug 20 2018

A343942 Number of even-length strict integer partitions of 2n+1.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 9, 13, 19, 27, 38, 52, 71, 96, 128, 170, 224, 292, 380, 491, 630, 805, 1024, 1295, 1632, 2049, 2560, 3189, 3959, 4896, 6038, 7424, 9100, 11125, 13565, 16496, 20013, 24223, 29249, 35244, 42378, 50849, 60896, 72789, 86841, 103424, 122960, 145937, 172928

Offset: 0

Gus Wiseman, Jun 09 2021

nonn

By conjugation, also the number of integer partitions of 2n+1 covering an initial interval of positive integers with greatest part even.

			The a(1) = 1 through a(7) = 13 strict partitions:
  (2,1)  (3,2)  (4,3)  (5,4)  (6,5)      (7,6)      (8,7)
         (4,1)  (5,2)  (6,3)  (7,4)      (8,5)      (9,6)
                (6,1)  (7,2)  (8,3)      (9,4)      (10,5)
                       (8,1)  (9,2)      (10,3)     (11,4)
                              (10,1)     (11,2)     (12,3)
                              (5,3,2,1)  (12,1)     (13,2)
                                         (5,4,3,1)  (14,1)
                                         (6,4,2,1)  (6,4,3,2)
                                         (7,3,2,1)  (6,5,3,1)
                                                    (7,4,3,1)
                                                    (7,5,2,1)
                                                    (8,4,2,1)
                                                    (9,3,2,1)

Ranked by A005117 (strict), A028260 (even length), and A300063 (odd sum).
Odd bisection of A067661 (non-strict: A027187).
The non-strict version is A236914.
The opposite type of strict partition (odd length and even sum) is A344650.
A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
A103919 counts partitions by sum and alternating sum (reverse: A344612).
A344610 counts partitions by sum and positive reverse-alternating sum.
Cf. A000070, A000097, A030229, A035294, A067659, A236559, A338907, A343941, A344649, A344654, A344739.

Table[Length[Select[IntegerPartitions[2n+1],UnsameQ@@#&&EvenQ[Length[#]]&]],{n,0,15}]

The Heinz numbers are A005117 /\ A028260 /\ A300063.

More terms from Bert Dobbelaere, Jun 12 2021

A351595 Number of odd-length integer partitions y of n such that y_i > y_{i+1} for all even i.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 2, 3, 4, 5, 6, 9, 10, 13, 16, 20, 24, 30, 35, 44, 52, 63, 74, 90, 105, 126, 148, 175, 204, 242, 280, 330, 382, 446, 515, 600, 690, 800, 919, 1060, 1214, 1398, 1595, 1830, 2086, 2384, 2711, 3092, 3506, 3988, 4516, 5122, 5788, 6552, 7388, 8345

Offset: 0

Gus Wiseman, Feb 25 2022

nonn

			The a(1) = 1 through a(12) = 10 partitions (A..C = 10..12):
  1   2   3   4   5     6     7     8     9     A     B       C
                  221   321   331   332   432   442   443     543
                              421   431   441   532   542     552
                                    521   531   541   551     642
                                          621   631   632     651
                                                721   641     732
                                                      731     741
                                                      821     831
                                                      33221   921
                                                              43221

The ordered version (compositions) is A000213 shifted right once.
All odd-length partitions are counted by A027193.
The opposite appears to be A122130, even-length A351008, any length A122129.
This appears to be the odd-length case of A122135, even-length A122134.
The case that is constant at odd indices:
- any length: A351005
- odd length: A351593
- even length: A035457
- opposite any length: A351006
- opposite odd length: A053251
- opposite even length: A351007
For equality instead of inequality:
- any length: A351003
- odd-length: A000009 (except at 0)
- even-length: A351012
- opposite any length: A351004
- opposite odd-length: A351594
- opposite even-length: A035363
Cf. A000041, A000070, A003242, A027383, A236559, A236914, A350842.

Table[Length[Select[IntegerPartitions[n],OddQ[Length[#]]&&And@@Table[#[[i]]>#[[i+1]],{i,2,Length[#]-1,2}]&]],{n,0,30}]

A352128 Number of strict integer partitions of n with (1) as many even parts as odd parts, and (2) as many even conjugate parts as odd conjugate parts.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 2, 0, 2, 2, 3, 0, 3, 0, 2, 2, 5, 2, 5, 4, 6, 7, 7, 8, 8, 9, 9, 13, 9, 14, 12, 20, 13, 25, 17, 33, 23, 40, 26, 50, 33, 59, 39, 68, 45, 84, 58, 92, 70, 115, 88, 132, 109, 156, 139, 182, 172, 212, 211

Offset: 0

Gus Wiseman, Mar 15 2022

nonn

			The a(n) strict partitions for selected n:
n = 3      18         22          28           31              32
   -----------------------------------------------------------------------
    (2,1)  (8,5,3,2)  (8,6,5,3)   (12,7,5,4)   (10,7,5,4,3,2)  (12,8,7,5)
           (8,6,3,1)  (8,7,5,2)   (12,8,5,3)   (10,7,6,5,2,1)  (12,9,7,4)
                      (12,7,2,1)  (12,9,5,2)   (10,8,5,4,3,1)  (16,9,4,3)
                                  (16,9,2,1)   (10,9,6,3,2,1)  (12,10,7,3)
                                  (12,10,5,1)                  (12,11,7,2)
                                                               (16,11,4,1)

The first condition is A239241, non-strict A045931 (ranked by A325698).
This is the strict version of A351977, ranked by A350946.
The second condition is A352129, non-strict A045931 (ranked by A350848).
A000041 counts integer partitions, strict A000009.
A130780 counts partitions with no more even than odd parts, strict A239243.
A171966 counts partitions with no more odd than even parts, strict A239240.
There are four statistics:
- A257991 = # of odd parts, conjugate A344616.
- A257992 = # of even parts, conjugate A350847.
There are four other pairings of statistics:
- A277579, strict A352131.
- A277103, ranked by A350944, strict A000700.
- A277579, ranked by A350943, strict A352130.
- A350948, ranked by A350945.
There are two other double-pairings of statistics:
- A351976, ranked by A350949.
- A351981, ranked by A351980.
The case of all four statistics equal is A351978, ranked by A350947.
Cf. A000070, A014105, A088218, A098123, A195017, A236559, A236914, A241638, A325700, A350839, A350941.

conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Count[#,?OddQ]==Count[#,?EvenQ]&&Count[conj[#],?OddQ]==Count[conj[#],?EvenQ]&]],{n,0,30}]

cp's OEIS Frontend

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A352129 Number of strict integer partitions of n with as many even conjugate parts as odd conjugate parts.

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A100824 Number of partitions of n with at most one odd part.

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A352130 Number of strict integer partitions of n with as many odd parts as even conjugate parts.

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A304620 Expansion of (1/(1 - x)) * Sum_{k>=0} x^(2*k) / Product_{j=1..2*k} (1 - x^j).

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A352131 Number of strict integer partitions of n with same number of even parts as odd conjugate parts.

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A306145 Expansion of (1/(1 - x)) * Sum_{k>=0} x^(2*k+1) / Product_{j=1..2*k+1} (1 - x^j).

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A343942 Number of even-length strict integer partitions of 2n+1.

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A304620 Expansion of (1/(1 - x)) * Sum_{k>=0} x^(2k) / Product_{j=1..2k} (1 - x^j).

A306145 Expansion of (1/(1 - x)) * Sum_{k>=0} x^(2k+1) / Product_{j=1..2k+1} (1 - x^j).