A239241 -id:A239241 - cp's OEIS frontend

A239261 Number of partitions of n having (sum of odd parts) = (sum of even parts).

1, 0, 0, 0, 1, 0, 0, 0, 4, 0, 0, 0, 12, 0, 0, 0, 30, 0, 0, 0, 70, 0, 0, 0, 165, 0, 0, 0, 330, 0, 0, 0, 704, 0, 0, 0, 1380, 0, 0, 0, 2688, 0, 0, 0, 4984, 0, 0, 0, 9394, 0, 0, 0, 16665, 0, 0, 0, 29970, 0, 0, 0, 52096, 0, 0, 0, 90090, 0, 0, 0, 152064, 0, 0, 0

Offset: 0

Clark Kimberling, Mar 13 2014

			a(8) counts these 4 partitions:  431, 41111, 3221, 221111.
From _Gus Wiseman_, Oct 24 2023: (Start)
The a(0) = 1 through a(12) = 12 partitions:
  ()  .  .  .  (211)  .  .  .  (431)     .  .  .  (633)
                               (3221)             (651)
                               (41111)            (4332)
                               (221111)           (5421)
                                                  (33222)
                                                  (52221)
                                                  (63111)
                                                  (432111)
                                                  (3222111)
                                                  (6111111)
                                                  (42111111)
                                                  (222111111)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..500

Cf. A239259, A239260, A239262, A239263, A000041.
The LHS (sum of odd parts) is counted by A113685.
The RHS (sum of even parts) is counted by A113686.
Without all the zeros we have a(4n) = A249914(n).
The strict case (without zeros) is A255001.
The Heinz numbers of these partitions are A366748, see also A019507.
A000009 counts partitions into odd parts, ranks A066208.
A035363 counts partitions into even parts, ranks A066207.
Cf. A028260, A045931, A106529, A239241, A241638, A325698, A365067.

z = 40; p[n_] := p[n] = IntegerPartitions[n]; f[t_] := f[t] = Length[t]
t1 = Table[f[Select[p[n], 2 Total[Select[#, OddQ]] < n &]], {n, z}] (* A239259 *)
t2 = Table[f[Select[p[n], 2 Total[Select[#, OddQ]] <= n &]], {n, z}] (* A239260 *)
t3 = Table[f[Select[p[n], 2 Total[Select[#, OddQ]] == n &]], {n, z}] (* A239261 *)
t4 = Table[f[Select[p[n], 2 Total[Select[#, OddQ]] > n &]], {n, z}] (* A239262 *)
t5 = Table[f[Select[p[n], 2 Total[Select[#, OddQ]] >= n &]], {n, z}] (* A239263 *)
(* Peter J. C. Moses, Mar 12 2014 *)

A239260(n) + a(n) + A239262(n) = A000041(n).
From David A. Corneth, Oct 25 2023: (Start)
a(4*n) = A000009(2*n) * A000041(n) for n >= 0.
a(4*n + r) = 0 for n >= 0 and r in {1, 2, 3}. (End)

More terms from Alois P. Heinz, Mar 15 2014

A240021 Number T(n,k) of partitions of n into distinct parts, where k is the difference between the number of odd parts and the number of even parts; triangle T(n,k), n>=0, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 0, 0, 1, 2, 1, 1, 1, 0, 1, 1, 1, 3, 1, 1, 1, 0, 2, 2, 2, 4, 1, 0, 1, 2, 1, 1, 4, 2, 4, 5, 1, 1, 1, 1, 2, 1, 2, 6, 3, 1, 6, 6, 1, 2, 2, 1, 3, 1, 5, 9, 3, 2, 9, 7, 2, 4, 3, 2, 3, 2, 8, 12, 4, 0, 1, 4, 12, 8, 3, 7, 4, 3, 4, 3, 14, 16, 4, 1, 1, 7, 16, 9, 6, 11, 5, 1, 4, 4, 6, 20, 20, 5, 2, 2

Offset: 0

Alois P. Heinz, Mar 31 2014

T(n,k) is defined for all n >= 0, k in A001057.  Row n contains all terms from the leftmost to the rightmost nonzero term.  All other terms (not in the triangle) are equal to 0. First nonzero term of column k>=0 is at n = k^2, first nonzero term of column k<=0 is at n = k*(k+1).
T(n,k) = T(n+k,-k).
T(2n*(2n+1),2n) = A000041(n).
T(4n^2+14n+11,2n+2) = A000070(n).
T(n^2,n) = 1.
T(n^2,n-1) = 0.
T(n^2,n-2) = A209815(n+1).
T(n^2+1,n-1) = A000065(n).
T(n,0) = A239241(n).
Sum_{k<=-1} T(n,k) = A239239(n).
Sum_{k<=0} T(n,k) = A239240(n).
Sum_{k>=1} T(n,k) = A239242(n).
Sum_{k>=0} T(n,k) = A239243(n).
Sum_{k=-1..1} T(n,k) = A239881(n).
T(n,-1) + T(n,1) = A239880(n).
Sum_{k=-n..n} T(n,k) = A000009 (row sums).

			T(12,-3) = 1: [6,4,2].
T(12,-2) = 2: [10,2], [8,4].
T(12,-1) = 1: [12].
T(12,0) = 2: [6,3,2,1], [5,4,2,1].
T(12,1) = 6: [9,2,1], [8,3,1], [7,4,1], [7,3,2], [6,5,1], [5,4,3].
T(12,2) = 3: [11,1], [9,3], [7,5].
T(13,-1) = 6: [10,2,1], [8,4,1], [8,3,2], [7,4,2], [6,5,2], [6,4,3].
T(14,-2) = 3: [12,2], [10,4], [8,6].
Triangle T(n,k) begins:
: n\k : -3 -2 -1  0  1  2  3  ...
+-----+--------------------------
:  0  :           1
:  1  :              1
:  2  :        1
:  3  :           1, 1
:  4  :        1, 0, 0, 1
:  5  :           2, 1
:  6  :     1, 1, 0, 1, 1
:  7  :        1, 3, 1
:  8  :     1, 1, 0, 2, 2
:  9  :        2, 4, 1, 0, 1
: 10  :     2, 1, 1, 4, 2
: 11  :        4, 5, 1, 1, 1
: 12  :  1, 2, 1, 2, 6, 3
: 13  :     1, 6, 6, 1, 2, 2
: 14  :  1, 3, 1, 5, 9, 3

Alois P. Heinz, Rows n = 0..500, flattened

Columns k=0-10 give: A239241, A239871(n+1), A240138, A240139, A240140, A240141, A240142, A240143, A240144, A240145, A240146.

Cf. A240009.

b:= proc(n, i) option remember; `if`(n>i*(i+1)/2, 0, `if`(n=0, 1,
      expand(b(n, i-1)+`if`(i>n, 0, b(n-i, i-1)*x^(2*irem(i, 2)-1)))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=ldegree(p)..degree(p)))(b(n$2)):
seq(T(n), n=0..20);

b[n_, i_] := b[n, i] = If[n>i*(i+1)/2, 0, If[n == 0, 1, Expand[b[n, i-1] + If[i>n, 0, b[n-i, i-1]*x^(2*Mod[i, 2]-1)]]]]; T[n_] := Function[{p}, Table[ Coefficient[p, x, i], {i, Exponent[p, x, Min], Exponent[p, x]}]][b[n, n]]; Table[ T[n], {n, 0, 20}] // Flatten (* Jean-François Alcover, Feb 11 2015, after Alois P. Heinz *)

N=20; q='q+O('q^N);
e(n) = if(n%2!=0, u, 1/u);
gf = prod(n=1,N, 1 + e(n)*q^n );
V = Vec( gf );
{ for (j=1, #V,  \\ print triangle, including leading zeros
    for (i=0, N-j, print1("   "));  \\ padding
    for (i=-j+1, j-1, print1(polcoeff(V[j], i, u),", "));
    print();
); }
/* Joerg Arndt, Apr 01 2014 */

G.f.: prod(n>=1, 1 + e(n)*q^n ) = 1 + sum(n>=1, e(n)*q^n * prod(k=1..n-1, 1+e(k)*q^k) ) where e(n) = u if n odd, otherwise 1/u; see Pari program. [Joerg Arndt, Apr 01 2014]

A350942 Number of odd parts minus number of even conjugate parts of the integer partition with Heinz number n.

Original entry on oeis.org

0, 1, 0, 1, 1, 0, 0, 3, -2, 1, 1, 2, 0, 0, -1, 3, 1, 0, 0, 3, -2, 1, 1, 2, -1, 0, 0, 2, 0, 1, 1, 5, -1, 1, -2, 0, 0, 0, -2, 3, 1, 0, 0, 3, 1, 1, 1, 4, -4, 1, -1, 2, 0, 0, -1, 2, -2, 0, 1, 1, 0, 1, 0, 5, -2, 1, 1, 3, -1, 0, 0, 2, 1, 0, 1, 2, -3, 0, 0, 5, -2, 1

Offset: 1

Gus Wiseman, Jan 28 2022

sign

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			First positions n such that a(n) = 6, 5, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5, -6, together with their prime indices, are:
  192: (2,1,1,1,1,1,1)
   32: (1,1,1,1,1)
   48: (2,1,1,1,1)
    8: (1,1,1)
   12: (2,1,1)
    2: (1)
    1: ()
   15: (3,2)
    9: (2,2)
   77: (5,4)
   49: (4,4)
  221: (7,6)
  169: (6,6)

The conjugate version is A350849.
This is a hybrid of A195017 and A350941.
Positions of 0's are A350943.
A000041 = integer partitions, strict A000009.
A056239 adds up prime indices, counted by A001222, row sums of A112798.
A122111 represents conjugation using Heinz numbers.
A257991 = # of odd parts, conjugate A344616.
A257992 = # of even parts, conjugate A350847.
A316524 = alternating sum of prime indices.
The following rank partitions:
  A325698: # of even parts = # of odd parts.
  A349157: # of even parts = # of odd conjugate parts, counted by A277579.
  A350848: # even conj parts = # odd conj parts, counted by A045931.
  A350943: # of even conjugate parts = # of odd parts, counted by A277579.
  A350944: # of odd parts = # of odd conjugate parts, counted by A277103.
  A350945: # of even parts = # of even conjugate parts, counted by A350948.
Cf. A026424, A028260, A130780, A171966, A239241, A241638, A325700, A350947, A350949, A350950, A350951.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[Count[primeMS[n],?OddQ]-Count[conj[primeMS[n]],?EvenQ],{n,100}]

A350947 Heinz numbers of integer partitions with the same number of even parts, odd parts, even conjugate parts, and odd conjugate parts.

Original entry on oeis.org

1, 6, 84, 210, 490, 525, 2184, 2340, 5460, 9464, 12012, 12740, 12870, 13650, 14625, 19152, 22308, 30030, 34125, 43940, 45144, 55770, 59150, 66066, 70070, 70785, 75075, 79625, 82992, 88920

Offset: 1

Gus Wiseman, Mar 14 2022

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The terms together with their prime indices begin:
      1: ()
      6: (2,1)
     84: (4,2,1,1)
    210: (4,3,2,1)
    490: (4,4,3,1)
    525: (4,3,3,2)
   2184: (6,4,2,1,1,1)
   2340: (6,3,2,2,1,1)
   5460: (6,4,3,2,1,1)
   9464: (6,6,4,1,1,1)
  12012: (6,5,4,2,1,1)
  12740: (6,4,4,3,1,1)
  12870: (6,5,3,2,2,1)
  13650: (6,4,3,3,2,1)
  14625: (6,3,3,3,2,2)
  19152: (8,4,2,2,1,1,1,1)
For example, the partition (6,6,4,1,1,1) has conjugate (6,3,3,3,2,2), and all four statistics are equal to 3, so 9464 is in the sequence.

These partitions are counted by A351978.
There are four individual statistics:
- A257991 counts odd parts, conjugate A344616.
- A257992 counts even parts, conjugate A350847.
There are six possible pairings of statistics:
- A325698: # of even parts = # of odd parts, counted by A045931.
- A349157: # of even parts = # of odd conjugate parts, counted by A277579.
- A350848: # of even conj parts = # of odd conj parts, counted by A045931.
- A350943: # of even conjugate parts = # of odd parts, counted by A277579.
- A350944: # of odd parts = # of odd conjugate parts, counted by A277103.
- A350945: # of even parts = # of even conjugate parts, counted by A350948.
There are three possible double-pairings of statistics:
- A350946, counted by A351977.
- A350949, counted by A351976.
- A351980, counted by A351981.
A056239 adds up prime indices, counted by A001222, row sums of A112798.
A122111 represents partition conjugation using Heinz numbers.
A195017 = # of even parts - # of odd parts.
A316524 = alternating sum of prime indices.
Cf. A026424, A028260, A098123, A239241, A241638, A325700, A350849, A350941, A350942, A350950, A350951.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Select[Range[1000],Count[primeMS[#],?EvenQ]==Count[primeMS[#],?OddQ]==Count[conj[primeMS[#]],?EvenQ]==Count[conj[primeMS[#]],?OddQ]&]

A257992(a(n)) = A257991(a(n)) = A350847(a(n)) = A344616(a(n)).

A350949 Heinz numbers of integer partitions with as many even parts as even conjugate parts and as many odd parts as odd conjugate parts.

Original entry on oeis.org

1, 2, 6, 9, 20, 30, 56, 75, 84, 125, 176, 210, 264, 294, 315, 350, 416, 441, 490, 525, 624, 660, 735, 924, 990, 1088, 1100, 1386, 1540, 1560, 1632, 1650, 1715, 2184, 2310, 2340, 2401, 2432, 2600, 3267, 3276, 3388, 3640, 3648, 3900, 4080, 4125, 5082, 5324, 5390

Offset: 1

Gus Wiseman, Mar 14 2022

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The terms together with their prime indices begin:
     1: ()
     2: (1)
     6: (2,1)
     9: (2,2)
    20: (3,1,1)
    30: (3,2,1)
    56: (4,1,1,1)
    75: (3,3,2)
    84: (4,2,1,1)
   125: (3,3,3)
   176: (5,1,1,1,1)
   210: (4,3,2,1)
   264: (5,2,1,1,1)
   294: (4,4,2,1)
   315: (4,3,2,2)
   350: (4,3,3,1)
   416: (6,1,1,1,1,1)

The second condition alone is A350944, counted by A277103.
The first condition alone is A350945, counted by A350948.
The case of all four statistics equal is A350947, counted by A351978.
These partitions are counted by A351976.
There are four other possible pairings of statistics:
- A045931: # even = # odd, ranked by A325698, strict A239241.
- A045931: # even conj = # odd conj, ranked by A350848, strict A352129.
- A277579: # even = # odd conj, ranked by A349157, strict A352131.
- A277579: # even conj = # odd, ranked by A350943, strict A352130.
There are two other possible double-pairings of statistics:
- A350946, counted by A351977.
- A351980, counted by A351981.
A056239 adds up prime indices, counted by A001222, row sums of A112798.
A122111 represents partition conjugation using Heinz numbers.
A195017 = # of even parts - # of odd parts.
A257991 counts odd parts, conjugate A344616.
A257992 counts even parts, conjugate A350847.
A316524 = alternating sum of prime indices.
Cf. A026424, A028260, A098123, A130780, A171966, A241638, A325700, A350849, A350941, A350942, A350950, A350951.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Select[Range[1000],Count[primeMS[#],?OddQ]==Count[conj[primeMS[#]],?OddQ]&&Count[primeMS[#],?EvenQ]==Count[conj[primeMS[#]],?EvenQ]&]

Intersection of A350944 and A350945.
A257991(a(n)) = A344616(a(n)).
A257992(a(n)) = A350847(a(n)).
Closed under A122111 (conjugation).

A098123 Number of compositions of n with equal number of even and odd parts.

Original entry on oeis.org

1, 0, 0, 2, 0, 4, 6, 6, 24, 28, 60, 130, 190, 432, 770, 1386, 2856, 5056, 9828, 18918, 34908, 68132, 128502, 244090, 470646, 890628, 1709136, 3271866, 6238986, 11986288, 22925630, 43932906, 84349336, 161625288, 310404768, 596009494

Offset: 0

Vladeta Jovovic, Sep 24 2004

			From _Gus Wiseman_, Jun 26 2022: (Start)
The a(0) = 1 through a(7) = 6 compositions (empty columns indicated by dots):
  ()  .  .  (12)  .  (14)  (1122)  (16)
            (21)     (23)  (1212)  (25)
                     (32)  (1221)  (34)
                     (41)  (2112)  (43)
                           (2121)  (52)
                           (2211)  (61)
(End)

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

For partitions: A045931, ranked by A325698, strict A239241 (conj A352129).
Column k=0 of A242498.
Without multiplicity: A242821, for partitions A241638 (ranked by A325700).
These compositions are ranked by A355321.
A047993 counts balanced partitions, ranked by A106529.
A108950/A108949 count partitions with more odd/even parts.
A130780/A171966 count partitions with more or as many odd/even parts.
Cf. A000700, A000712, A001405, A026424, A027193, A028260, A097613, A240009, A277579 (ranked by A349157).
Cf. A025178.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Count[#,?EvenQ]==Count[#,?OddQ]&]],{n,0,15}] (* Gus Wiseman, Jun 26 2022 *)

a(n) = Sum_{k=floor(n/3)..floor(n/2)} C(2*n-4*k,n-2*k)*C(n-1-k,2*n-4*k-1).
Recurrence: n*(2*n-7)*a(n) = 2*(n-2)*(2*n-5)*a(n-2) + 2*(2*n-7)*(2*n-3)*a(n-3) - (n-4)*(2*n-3)*a(n-4). - Vaclav Kotesovec, May 01 2014
a(n) ~ sqrt(c) * d^n / sqrt(Pi*n), where d = 1.94696532812840456026081823863... is the root of the equation 1-4*d-2*d^2+d^4 = 0, c = 0.225563290820392765554898545739... is the root of the equation 43*c^4-18*c^2+8*c-1=0. - Vaclav Kotesovec, May 01 2014
G.f.: 1/sqrt(1 - 4*x^3/(1-x^2)^2). - Seiichi Manyama, May 01 2025

A026010 a(n) = number of (s(0), s(1), ..., s(n)) such that s(i) is a nonnegative integer and |s(i) - s(i-1)| = 1 for i = 1,2,...,n and s(0) = 2. Also a(n) = sum of numbers in row n+1 of array T defined in A026009.

Original entry on oeis.org

1, 2, 4, 7, 14, 25, 50, 91, 182, 336, 672, 1254, 2508, 4719, 9438, 17875, 35750, 68068, 136136, 260338, 520676, 999362, 1998724, 3848222, 7696444, 14858000, 29716000, 57500460, 115000920, 222981435, 445962870, 866262915, 1732525830, 3370764540

Offset: 0

Clark Kimberling

nonn

Conjecture: a(n) is the number of integer compositions of n + 2 in which the even parts appear as often at even positions as at odd positions (confirmed up to n = 19). - Gus Wiseman, Mar 17 2018

			The a(3) = 7 compositions of 5 in which the even parts appear as often at even positions as at odd positions are (5), (311), (131), (113), (221), (122), (11111). Missing are (41), (14), (32), (23), (212), (2111), (1211), (1121), (1112). - _Gus Wiseman_, Mar 17 2018

Michael De Vlieger, Table of n, a(n) for n = 0..3326
Christian Krattenthaler, Daniel Yaqubi, Some determinants of path generating functions, II, arXiv:1802.05990 [math.CO], 2018; Adv. Appl. Math. 101 (2018), 232-265.

First differences of A050168. Pairwise sums of A037952.

Cf. A000712, A001405, A005774, A045931, A063886, A097613, A130780, A171966, A239241, A299926, A300061, A300787, A300788, A300789.

[(&+[Binomial(Floor((n+k)/2), Floor(k/2)): k in [0..n]]): n in [0..40]]; // G. C. Greubel, Nov 08 2018

Array[Sum[Binomial[Floor[(# + k)/2], Floor[k/2]], {k, 0, #}] &, 34, 0] (* Michael De Vlieger, May 16 2018 *)
Table[2^(-1 + n)*(((2 + 3*#)*Gamma[(1 + #)/2])/(Sqrt[Pi]*Gamma[2 + #/2]) &[n + Mod[n, 2]]), {n,0,40}] (* Peter Pein, Nov 08 2018 *)
Table[(1/2)^((5 - (-1)^n)/2)*(6*n + 7 - 3*(-1)^n)*CatalanNumber[(2*n + 1 - (-1)^n)/4], {n, 0, 40}] (* G. C. Greubel, Nov 08 2018 *)

vector(40, n, n--; sum(k=0,n, binomial(floor((n+k)/2), floor(k/2)))) \\ G. C. Greubel, Nov 08 2018

a(2*n) = ((3*n + 1)/(2*n + 1))*C(2*n + 1, n)= A051924(1+n), n>=0, a(2*n-1) = a(2*n)/2 = A097613(1+n), n >= 1. - Herbert Kociemba, May 08 2004
a(n) = Sum_{k=0..n} binomial(floor((n+k)/2), floor(k/2)). - Paul Barry, Jul 15 2004
Inverse binomial transform of A005774: (1, 3, 9, 26, 75, 216, ...). - Gary W. Adamson, Oct 22 2007
Conjecture: (n+3)*a(n) - 2*a(n-1) + (-5*n-3)*a(n-2) + 2*a(n-3) + 4*(n-3)*a(n-4) = 0. - R. J. Mathar, Jun 20 2013
a(n) = (1/2)^((5 - (-1)^n)/2)*(6*n + 7 - 3*(-1)^n)*Catalan((2*n + 1 - (-1)^n)/4), where Catalan is the Catalan number = A000108. - G. C. Greubel, Nov 08 2018

A350849 Number of odd conjugate parts minus number of even parts in the integer partition with Heinz number n.

Original entry on oeis.org

0, 1, 1, 0, 3, 0, 3, 1, -2, 2, 5, 1, 5, 2, 0, 0, 7, -1, 7, 3, 0, 4, 9, 0, 0, 4, -1, 3, 9, 1, 11, 1, 2, 6, 0, -2, 11, 6, 2, 2, 13, 1, 13, 5, 1, 8, 15, 1, -2, 1, 4, 5, 15, -2, 2, 2, 4, 8, 17, 0, 17, 10, 1, 0, 2, 3, 19, 7, 6, 1, 19, -1, 21, 10, 1, 7, 0, 3, 21, 3

Offset: 1

Gus Wiseman, Jan 28 2022

sign

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			First positions n such that a(n) = 4, 3, 2, 1, 0, -1, -2, -3, -4, together with their prime indices, are:
   22: (5,1)
    5: (3)
   10: (3,1)
    2: (1)
    1: ()
   18: (2,2,1)
    9: (2,2)
  162: (2,2,2,2,1)
   81: (2,2,2,2)

This is a hybrid of A195017 and A350941.
Positions of 0's are A349157.
Counting even conjugate parts instead of even parts gives A350941.
The conjugate version is A350942.
A257991 counts odd parts, conjugate A344616.
A257992 counts even parts, conjugate A350847.
The following rank partitions:
  A325698: # of even parts = # of odd parts.
  A349157: # of even parts = # of odd conjugate parts, counted by A277579.
  A350848: # even conj parts = # odd conj parts, counted by A045931.
  A350943: # of even conjugate parts = # of odd parts, counted by A277579.
  A350944: # of odd parts = # of odd conjugate parts, counted by A277103.
  A350945: # of even parts = # of even conjugate parts, counted by A350948.
A000041 = integer partitions, strict A000009.
A056239 adds up prime indices, counted by A001222, row sums of A112798.
A122111 represents conjugation using Heinz numbers.
A316524 = alternating sum of prime indices.
Cf. A026424, A028260, A130780, A171966, A239241, A241638, A325700, A350947, A350949, A350950, A350951.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[Count[conj[primeMS[n]],?OddQ]-Count[primeMS[n],?EvenQ],{n,100}]

a(n) = A344616(n) - A257992(n).

A350946 Heinz numbers of integer partitions with as many even parts as odd parts and as many even conjugate parts as odd conjugate parts.

Original entry on oeis.org

1, 6, 65, 84, 210, 216, 319, 490, 525, 532, 731, 1254, 1403, 1924, 2184, 2340, 2449, 2470, 3024, 3135, 3325, 3774, 4028, 4141, 4522, 5311, 5460, 7030, 7314, 7315, 7560, 7776, 7942, 8201, 8236, 9048, 9435, 9464, 10659, 10921, 11484, 11914, 12012, 12025, 12740

Offset: 1

Gus Wiseman, Mar 14 2022

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The terms together with their prime indices begin:
     1: ()
     6: (2,1)
    65: (6,3)
    84: (4,2,1,1)
   210: (4,3,2,1)
   216: (2,2,2,1,1,1)
   319: (10,5)
   490: (4,4,3,1)
   525: (4,3,3,2)
   532: (8,4,1,1)
   731: (14,7)
  1254: (8,5,2,1)
  1403: (18,9)
  1924: (12,6,1,1)
  2184: (6,4,2,1,1,1)
  2340: (6,3,2,2,1,1)
  2449: (22,11)
  2470: (8,6,3,1)
For example, the prime indices of 532 are (8,4,1,1), even/odd counts 2/2, and the prime indices of the conjugate 3024 are (4,2,2,2,1,1,1,1), with even/odd counts 4/4; so 532 belongs to the sequence.

For the first condition alone:
- counted by A045931 (strict A239241)
- ordered version (compositions) A098123
- ranked by A325698
- without multiplicity A325700 (counted by A241638)
The second condition alone is ranked by A350848, strict A352129.
These partitions are counted by A351977, strict A352128.
There are four statistics:
- A257991 = # of odd parts, conjugate A344616.
- A257992 = # of even parts, conjugate A350847.
There are four other possible pairings of statistics:
- A349157: # of even parts = # of odd conjugate parts, counted by A277579.
- A350943: # of even conj parts = # of odd parts, strict counted by A352130.
- A350944: # of odd parts = # of odd conjugate parts, counted by A277103.
- A350945: # of even parts = # of even conjugate parts, counted by A350948.
There are two other possible double-pairings of statistics:
- A350949, counted by A351976.
- A351980, counted by A351981.
The case of all four statistics equal is A350947, counted by A351978.
A056239 adds up prime indices, counted by A001222, row sums of A112798.
A122111 represents partition conjugation using Heinz numbers.
A195017 = # of even parts - # of odd parts.
A316524 = alternating sum of prime indices.
Cf. A026424, A028260, A130780, A171966, A347450, A350849, A350941, A350942, A350950, A350951.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Select[Range[1000],#==1||Mean[Mod[primeMS[#],2]]== Mean[Mod[conj[primeMS[#]],2]]==1/2&]

Closed under A122111 (conjugation).
Intersection of A325698 and A350848.
A257992(a(n)) = A257991(a(n)).
A350847(a(n)) = A344616(a(n)).

A350950 Number of even parts minus number of even conjugate parts in the integer partition with Heinz number n.

Original entry on oeis.org

0, 0, 1, -1, 0, 0, 1, 0, 0, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, 0, 1, 0, 0, -3, 0, 3, 1, 1, 0, 0, 0, -1, -1, -2, 0, 1, 0, 0, -1, 0, 1, 1, 0, 2, -1, 0, 1, -2, -2, -1, 1, 1, 2, -3, 0, 0, 0, 0, -1, 1, -1, 3, -1, -2, 0, 0, 0, -1, -1, 1, 1, 0, 0, 0, 1, -3, 1, 1, 0

Offset: 1

Gus Wiseman, Mar 14 2022

sign

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The prime indices of 78 are (6,2,1), with conjugate (3,2,1,1,1,1), so a(78) = 2 - 1 = 1.

The version comparing even with odd parts is A195017.
The version comparing even with odd conjugate parts is A350849.
The version comparing even conjugate with odd conjugate parts is A350941.
The version comparing odd with even conjugate parts is A350942.
Positions of 0's are A350945, counted by A350948.
The version comparing odd with odd conjugate parts is A350951.
There are four individual statistics:
- A257991 counts odd parts, conjugate A344616.
- A257992 counts even parts, conjugate A350847.
There are five other possible pairings of statistics:
- A325698: # of even parts = # of odd parts, counted by A045931.
- A349157: # of even parts = # of odd conjugate parts, counted by A277579.
- A350848: # of even conj parts = # of odd conj parts, counted by A045931.
- A350943: # of even conjugate parts = # of odd parts, counted by A277579.
- A350944: # of odd parts = # of odd conjugate parts, counted by A277103.
There are three possible double-pairings of statistics:
- A350946, counted by A351977.
- A350949, counted by A351976.
- A351980, counted by A351981.
The case of all four statistics equal is A350947, counted by A351978.
A056239 adds up prime indices, counted by A001222, row sums of A112798.
A116482 counts partitions by number of even parts.
A122111 represents partition conjugation using Heinz numbers.
A316524 gives the alternating sum of prime indices.
Cf. A028260, A088218, A098123, A236559, A239241, A241638, A325700, A347450.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[Count[primeMS[n],?EvenQ]-Count[conj[primeMS[n]],?EvenQ],{n,100}]

a(n) = A257992(n) - A350847(n).

a(A122111(n)) = -a(n), where A122111 represents partition conjugation.

cp's OEIS Frontend

A239261 Number of partitions of n having (sum of odd parts) = (sum of even parts).

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A240021 Number T(n,k) of partitions of n into distinct parts, where k is the difference between the number of odd parts and the number of even parts; triangle T(n,k), n>=0, read by rows.

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A350942 Number of odd parts minus number of even conjugate parts of the integer partition with Heinz number n.

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A350947 Heinz numbers of integer partitions with the same number of even parts, odd parts, even conjugate parts, and odd conjugate parts.

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A350949 Heinz numbers of integer partitions with as many even parts as even conjugate parts and as many odd parts as odd conjugate parts.

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A098123 Number of compositions of n with equal number of even and odd parts.

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A026010 a(n) = number of (s(0), s(1), ..., s(n)) such that s(i) is a nonnegative integer and |s(i) - s(i-1)| = 1 for i = 1,2,...,n and s(0) = 2. Also a(n) = sum of numbers in row n+1 of array T defined in A026009.

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A350849 Number of odd conjugate parts minus number of even parts in the integer partition with Heinz number n.

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