A239241 -id:A239241 - cp's OEIS frontend

A239243 Number of partitions of n into distinct parts for which (number of odd parts) >= (number of even parts).

1, 1, 0, 2, 1, 3, 2, 4, 4, 6, 7, 8, 11, 11, 17, 16, 25, 22, 36, 31, 49, 44, 68, 61, 90, 85, 120, 118, 156, 160, 204, 217, 261, 291, 337, 386, 429, 507, 548, 662, 694, 854, 882, 1096, 1112, 1396, 1406, 1765, 1768, 2219, 2223, 2776, 2784, 3451, 3484, 4275

Offset: 0

Clark Kimberling, Mar 13 2014

a(n) = Sum_{k>=0} A240021(n,k). - Alois P. Heinz, Apr 02 2014

			a(8) = 4 counts these partitions:  71, 53, 521, 431.

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

Cf. A239239, A239240, A239241, A239242, A000009.

b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2, 0,
     `if`(n=0, `if`(t>=0, 1, 0 ), b(n, i-1, t)+`if`(i>n, 0,
      b(n-i, i-1, t+`if`(irem(i, 2)=1, 1, -1)))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..60);  # Alois P. Heinz, Mar 15 2014

z = 55; p[n_] := p[n] = IntegerPartitions[n]; d[u_] := d[u] = DeleteDuplicates[u]; g[u_] := g[u] = Length[u];
Table[g[Select[Select[p[n], d[#] == # &], Count[#, ?OddQ] < Count[#, ?EvenQ] &]], {n, 0, z}] (* A239239 *)
Table[g[Select[Select[p[n], d[#] == # &], Count[#, ?OddQ] <= Count[#, ?EvenQ] &]], {n, 0, z}] (* A239240 *)
Table[g[Select[Select[p[n], d[#] == # &], Count[#, ?OddQ] == Count[#, ?EvenQ] &]], {n, 0, z}] (* A239241 *)
Table[g[Select[Select[p[n], d[#] == # &], Count[#, ?OddQ] > Count[#, ?EvenQ] &]], {n, 0, z}] (* A239242 *)
Table[g[Select[Select[p[n], d[#] == # &], Count[#, ?OddQ] >= Count[#, ?EvenQ] &]], {n, 0, z}] (* A239243 *)
(* Peter J. C. Moses, Mar 10 2014 *)
b[n_, i_, t_] := b[n, i, t] = If[n>i*(i+1)/2, 0, If[n==0, If[t>=0, 1, 0], b[n, i-1, t]+If[i>n, 0, b[n-i, i-1, t+If[Mod[i, 2]==1, 1, -1]]]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Aug 30 2016, after Alois P. Heinz *)

a(n) + A239239(n) = A000009(n) for n >=1.

A019507 Droll numbers: numbers > 1 whose sum of even prime factors equals the sum of odd prime factors.

Original entry on oeis.org

72, 240, 672, 800, 2240, 4224, 5184, 6272, 9984, 14080, 17280, 33280, 39424, 48384, 52224, 57600, 93184, 116736, 161280, 174080, 192000, 247808, 304128, 373248, 389120, 451584, 487424, 537600, 565248, 585728, 640000, 718848, 1013760, 1089536, 1244160, 1384448

Offset: 1

Mario Velucchi (mathchess(AT)velucchi.it)

nonn

			6272 = 2*2*2*2*2*2*2*7*7 is droll since 2+2+2+2+2+2+2 = 14 = 7+7.

Donovan Johnson, Table of n, a(n) for n = 1..1000
Giovanni Resta, Droll numbers, Numbers Aplenty.

For count instead of sum we have A072978.
Partitions of this type are counted by A239261, without zero terms A249914.
For prime indices instead of factors we have A366748, zeros of A366749.
The LHS is A366839 with alternating zeros, for indices A366531, triangle A113686.
The RHS is A366840, for indices A366528, triangle A113685.
A000009 counts partitions into odd parts, ranks A066208.
A035363 counts partitions into even parts, ranks A066207.
A112798 lists prime indices, length A001222, sum A056239.
A257991 counts odd prime indices, even A257992.
A300061 lists numbers with even sum of prime indices, odd A300063.
Cf. A028260, A045931, A100367, A106529, A195017, A239241, A241638, A325698, A325700, A366533.

f:= proc(k, m) # numbers whose sum of prime factors >= m is k; m is prime
   local S,p,j;
   option remember;
   if k = 0 then return [1]
   elif m > k then return []
   fi;
   S:= NULL:
   p:= nextprime(m);
   for j from k by -m to 0 do
     S:= S, op(map(`*`,  procname(j,p) , m^((k-j)/m)))
   od;
   [S]
end proc:
g:= proc(N) local m,R;
  R:= NULL;
  for m from 1 while 2^m < N do
   R:= R, op(map(`*`,select(`<=`,f(2*m,3), N/2^m),2^m));
  od;
  sort([R])
end proc:
g(10^8); # Robert Israel, Feb 20 2025

Select[Range[2, 2*10^6, 2], First[#] == Total[Rest[#]] & [Times @@@ FactorInteger[#]] &] (* Paolo Xausa, Feb 19 2025 *)

isok(n) = {if (n % 2, return (0)); f = factor(n); return (2*f[1,2] == sum(i=2, #f~, f[i,1]*f[i,2]));} \\ Michel Marcus, Jun 21 2013

These are even numbers k such that A366839(k/2) = A366840(k). - Gus Wiseman, Oct 25 2023 (corrected Feb 19 2025)

Name edited by Paolo Xausa, Feb 19 2025

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}]

A239242 Number of partitions of n into distinct parts for which (number of odd parts) > (number of even parts).

Original entry on oeis.org

0, 1, 0, 1, 1, 1, 2, 1, 4, 2, 6, 3, 9, 5, 12, 9, 17, 14, 22, 22, 29, 33, 38, 48, 50, 68, 65, 95, 86, 128, 113, 172, 149, 226, 197, 295, 260, 379, 342, 485, 449, 613, 587, 773, 762, 967, 987, 1206, 1269, 1497, 1623, 1855, 2063, 2289, 2610, 2823, 3280, 3471

Offset: 0

Clark Kimberling, Mar 13 2014

a(n) = Sum_{k>=1} A240021(n,k). - Alois P. Heinz, Apr 02 2014

			a(8) = 4 counts these partitions:  71, 53, 521, 431.

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

Cf. A239239, A239240, A239241, A239243, A000009.

b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2, 0,
     `if`(n=0, `if`(t>0, 1, 0 ), b(n, i-1, t)+`if`(i>n, 0,
      b(n-i, i-1, t+`if`(irem(i, 2)=1, 1, -1)))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..60);  # Alois P. Heinz, Mar 15 2014

z = 55; p[n_] := p[n] = IntegerPartitions[n]; d[u_] := d[u] = DeleteDuplicates[u]; g[u_] := g[u] = Length[u];
Table[g[Select[Select[p[n], d[#] == # &], Count[#, ?OddQ] < Count[#, ?EvenQ] &]], {n, 0, z}] (* A239239 *)
Table[g[Select[Select[p[n], d[#] == # &], Count[#, ?OddQ] <= Count[#, ?EvenQ] &]], {n, 0, z}] (* A239240 *)
Table[g[Select[Select[p[n], d[#] == # &], Count[#, ?OddQ] == Count[#, ?EvenQ] &]], {n, 0, z}] (* A239241 *)
Table[g[Select[Select[p[n], d[#] == # &], Count[#, ?OddQ] > Count[#, ?EvenQ] &]], {n, 0, z}] (* A239242 *)
Table[g[Select[Select[p[n], d[#] == # &], Count[#, ?OddQ] >= Count[#, ?EvenQ] &]], {n, 0, z}] (* A239243 *)
(* Peter J. C. Moses, Mar 10 2014 *)
b[n_, i_, t_] := b[n, i, t] = If[n>i*(i+1)/2, 0, If[n==0, If[t>0, 1, 0], b[n, i-1, t]+If[i>n, 0, b[n-i, i-1, t+If[Mod[i, 2]==1, 1, -1]]]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Aug 30 2016, after Alois P. Heinz *)

a(n) + A239240(n) = A000009(n) for n >=1.

A239871 Number of strict partitions of n having 1 more even part than odd, so that there is at least one ordering of the parts in which the even and odd parts alternate, and the first and last terms are even.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 1, 1, 1, 2, 1, 4, 1, 6, 1, 9, 2, 12, 3, 16, 6, 20, 10, 25, 17, 30, 26, 36, 40, 43, 57, 51, 81, 61, 110, 74, 148, 91, 193, 113, 250, 144, 316, 184, 397, 239, 491, 311, 603, 407, 732, 530, 885, 692, 1061, 895, 1268, 1155, 1508, 1478, 1790

Offset: 0

Clark Kimberling, Mar 29 2014

Let c(n) be the number of strict partitions (that is, every part has multiplicity 1) of n having 1 more odd part than even, so that there is an ordering of parts for which the odd and even parts alternate and the first and last terms are odd. Then c(n) = a(n+1) for n >= 0.

			a(11) counts these 4 partitions:  812, 614, 632, 452.

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

Cf. A239241, A239872, A239873, A239832.

Column k=-1 of A240021.

b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2 or
      abs(t)>n, 0, `if`(n=0, 1, b(n, i-1, t)+
      `if`(i>n, 0, b(n-i, i-1, t+(2*irem(i, 2)-1)))))
    end:
a:= n-> b(n$2, 1):
seq(a(n), n=0..80);  # Alois P. Heinz, Apr 02 2014

d[n_] := Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &];
p[n_] := p[n] = Select[d[n], Count[#, ?OddQ] == -1 + Count[#, ?EvenQ] &]; t =  Table[p[n], {n, 0, 20}]
TableForm[t] (* shows the partitions *)
u = Table[Length[p[n]], {n, 0, 70}]  (* A239871 *)
(* Peter J. C. Moses, Mar 10 2014 *)
b[n_, i_, t_] := b[n, i, t] = If[n > i*(i + 1)/2 || Abs[t] > n, 0, If[n == 0, 1, b[n, i - 1, t] + If[i > n, 0, b[n - i, i - 1, t + (2*Mod[i, 2] - 1)]]]]; a[n_] := b[n, n, 1]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Nov 16 2015, after Alois P. Heinz *)

a(n) = [x^n y^(-1)] Product_{i>=1} 1+x^i*y^(2*(i mod 2)-1). - Alois P. Heinz, Apr 03 2014

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}]

A239239 Number of strict partitions of n having fewer odd parts than even.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 2, 1, 2, 2, 3, 4, 4, 7, 5, 11, 7, 16, 10, 23, 15, 32, 21, 43, 32, 57, 45, 74, 66, 96, 92, 123, 129, 157, 175, 199, 239, 253, 316, 320, 419, 406, 544, 514, 704, 652, 898, 825, 1142, 1045, 1435, 1321, 1798, 1669, 2234, 2103, 2766, 2646, 3404

Offset: 0

Clark Kimberling, Mar 13 2014

a(n) = Sum_{k<=-1} A240021(n,k). - Alois P. Heinz, Apr 02 2014

			a(6) counts these partitions:  6, 42.

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

Cf. A239240, A239241, A239242, A239243, A000009.

b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2, 0,
     `if`(n=0, `if`(t<0, 1, 0 ), b(n, i-1, t)+`if`(i>n, 0,
      b(n-i, i-1, t+`if`(irem(i, 2)=1, 1, -1)))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..60);  # Alois P. Heinz, Mar 15 2014

z = 55; p[n_] := p[n] = IntegerPartitions[n]; d[u_] := d[u] = DeleteDuplicates[u]; g[u_] := g[u] = Length[u];
Table[g[Select[Select[p[n], d[#] == # &], Count[#, ?OddQ] < Count[#, ?EvenQ] &]], {n, 0, z}] (* A239239 *)
Table[g[Select[Select[p[n], d[#] == # &], Count[#, ?OddQ] <= Count[#, ?EvenQ] &]], {n, 0, z}] (* A239240 *)
Table[g[Select[Select[p[n], d[#] == # &], Count[#, ?OddQ] == Count[#, ?EvenQ] &]], {n, 0, z}] (* A239241 *)
Table[g[Select[Select[p[n], d[#] == # &], Count[#, ?OddQ] > Count[#, ?EvenQ] &]], {n, 0, z}] (* A239242 *)
Table[g[Select[Select[p[n], d[#] == # &], Count[#, ?OddQ] >= Count[#, ?EvenQ] &]], {n, 0, z}] (* A239243 *)
(* Peter J. C. Moses, Mar 10 2014 *)
b[n_, i_, t_] := b[n, i, t] = If[n>i*(i+1)/2, 0, If[n == 0, If[t<0, 1, 0], b[n, i-1, t] + If[i>n, 0, b[n-i, i-1, t+If[Mod[i, 2] == 1, 1, -1]]]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Aug 29 2016, after Alois P. Heinz *)

a(n) + A239243(n) = A000009(n) for n >=1.

A239872 Number of strict partitions of 2n having 1 more even part than odd, so that there is at least one ordering of the parts in which the even and odd parts alternate, and the first and last terms are even.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 2, 3, 6, 10, 17, 26, 40, 57, 81, 110, 148, 193, 250, 316, 397, 491, 603, 732, 885, 1061, 1268, 1508, 1790, 2120, 2510, 2970, 3517, 4170, 4950, 5887, 7013, 8371, 10005, 11979, 14353, 17217, 20654, 24785, 29725, 35637, 42672, 51046, 60962

Offset: 0

Clark Kimberling, Mar 29 2014

Let c(n) be the number of strict partitions (that is, every part has multiplicity 1) of 2n having 1 more even part than odd, so that there is an ordering of parts for which the even and odd parts alternate and the first and last terms are even. This sequence is nondecreasing, unlike A239871, of which it is a bisection; the other bisection is A239873.

			a(9) counts these 3 partitions of 18:  [18], [8,3,4,1,2], [6,5,4,1,2].

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

Cf. A239241, A239871, A239873, A239832.

b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2 or
      abs(t)-n>0, 0, `if`(n=0, 1, b(n, i-1, t)+
      `if`(i>n, 0, b(n-i, i-1, t+(2*irem(i, 2)-1)))))
    end:
a:= n-> b(2*n$2, 1):
seq(a(n), n=0..60);  # Alois P. Heinz, Apr 01 2014

d[n_] := Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &]; p[n_] := p[n] = Select[d[n], Count[#, ?OddQ] == -1 + Count[#, ?EvenQ] &]; t = Table[p[n], {n, 0, 20}]
TableForm[t] (* shows the partitions *)
u = Table[Length[p[2 n]], {n, 0, 40}]  (* A239872 *)
(* Peter J. C. Moses, Mar 10 2014 *)
b[n_, i_, t_] := b[n, i, t] = If[n > i*(i+1)/2 || Abs[t]-n > 0, 0, If[n == 0, 1, b[n, i-1, t] + If[i>n, 0, b[n-i, i-1, t + (2*Mod[i, 2] - 1)]]]]; a[n_] := b[2*n, 2*n, 1]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Oct 28 2015, after Alois P. Heinz *)

A300787 Number of integer partitions of n in which the even parts appear as often at even positions as at odd positions.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 6, 8, 12, 15, 21, 27, 38, 47, 63, 79, 106, 130, 170, 209, 272, 330, 422, 512, 653, 784, 986, 1183, 1482, 1765, 2191, 2604, 3218, 3804, 4666, 5504, 6726, 7898, 9592, 11240, 13602, 15880, 19122, 22277, 26733, 31048, 37102, 43003, 51232, 59220

Offset: 0

Gus Wiseman, Mar 12 2018

nonn

			The a(7) = 8 partitions: (7), (511), (421), (331), (322), (31111), (22111), (1111111). Missing are: (61), (52), (43), (4111), (3211), (2221), (211111).

Even- and odd-indexed terms are A006330 and A001523 respectively, which add up to A000712.

Cf. A000898, A001405, A026010, A045931, A058696, A063886, A097613, A130780, A171966, A239241, A300788, A300789.

cobal[y_]:=Sum[(-1)^x,{x,Join@@Position[y,_?EvenQ]}];
Table[Length[Select[IntegerPartitions[n],cobal[#]===0&]],{n,0,50}]

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

A239243 Number of partitions of n into distinct parts for which (number of odd parts) >= (number of even parts).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A019507 Droll numbers: numbers > 1 whose sum of even prime factors equals the sum of odd prime factors.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A239242 Number of partitions of n into distinct parts for which (number of odd parts) > (number of even parts).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A239871 Number of strict partitions of n having 1 more even part than odd, so that there is at least one ordering of the parts in which the even and odd parts alternate, and the first and last terms are even.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

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

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A239239 Number of strict partitions of n having fewer odd parts than even.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A239872 Number of strict partitions of 2n having 1 more even part than odd, so that there is at least one ordering of the parts in which the even and odd parts alternate, and the first and last terms are even.

Views