A108949 -id:A108949 - cp's OEIS frontend

A338860 The excess of the number of partitions of n with more odd parts than even parts over the number of partitions of n with more even parts than odd parts.

0, 1, 0, 2, 1, 3, 4, 6, 8, 11, 17, 21, 30, 38, 53, 68, 90, 115, 150, 192, 243, 312, 390, 496, 613, 775, 951, 1193, 1456, 1810, 2200, 2715, 3285, 4026, 4856, 5909, 7106, 8595, 10301, 12394, 14809, 17728, 21118, 25171, 29891, 35489, 42018, 49702, 58678, 69180

Offset: 0

Jeremy Lovejoy, Jan 12 2021

nonn

			The 3 partitions of 4 with more odd parts than even parts are [3,1], [2,1,1], and [1,1,1,1], while the 2 partitions of 4 with more even parts than odd parts are [4] and [2,2].   Hence a(4) = 3-2 = 1.

Alois P. Heinz, Table of n, a(n) for n = 0..2000
B. Kim, E. Kim, and J. Lovejoy, Parity bias in partitions, European J. Combin., 89 (2020), 103159, 19 pp.

Cf. A108949, A108950.

b:= proc(n, i, t) option remember; `if`(n=0, signum(t), `if`(i<1, 0,
      b(n, i-1, t)+ b(n-i, min(n-i, i), t+(2*irem(i, 2)-1))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..55);  # Alois P. Heinz, Jan 14 2021

b[n_, i_, t_] := b[n, i, t] = If[n == 0, Sign[t], If[i < 1, 0,
   b[n, i-1, t] + b[n-i, Min[n-i, i], t + (2*Mod[i, 2]-1)]]];
a[n_] := b[n, n, 0];
Table[a[n], {n, 0, 55}] (* Jean-François Alcover, Sep 09 2022, after Alois P. Heinz *)

for(n=0,43,my(me=0,mo=0);forpart(v=n,my(x=Vec(v),se=sum(k=1,#x,x[k]%2==0),so=sum(k=1,#x,x[k]%2>0));me+=(se>so);mo+=(so>se));print1(mo-me,", ")) \\ Hugo Pfoertner, Jan 13 2021

G.f.: (Product_{k>=1} 1/(1-x^(2*k-1)))*Sum_{n>=1} q^(2*n^2-n)*(1-q^n)/Product_{k=1..n} (1-q^(2*k))^2.

a(n) = A108950(n) - A108949(n).

A355321 Numbers k such that the k-th composition in standard order has the same number of even parts as odd.

Original entry on oeis.org

0, 5, 6, 17, 18, 20, 24, 43, 45, 46, 53, 54, 58, 65, 66, 68, 72, 80, 96, 139, 141, 142, 149, 150, 154, 163, 165, 166, 169, 172, 177, 178, 180, 184, 197, 198, 202, 209, 210, 212, 216, 226, 232, 257, 258, 260, 264, 272, 288, 320, 343, 347, 349, 350, 363, 365

Offset: 1

Gus Wiseman, Jun 28 2022

nonn

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The terms together with their corresponding compositions begin:
   0: ()
   5: (2,1)
   6: (1,2)
  17: (4,1)
  18: (3,2)
  20: (2,3)
  24: (1,4)
  43: (2,2,1,1)
  45: (2,1,2,1)
  46: (2,1,1,2)
  53: (1,2,2,1)
  54: (1,2,1,2)
  58: (1,1,2,2)
  65: (6,1)
  66: (5,2)
  68: (4,3)
  72: (3,4)
  80: (2,5)
  96: (1,6)

A subset of A001969 (evil numbers), complement A000069.
These compositions are counted by A098123, without multiplicity A242821.
The version for partitions is A325698, counted by A045931.
For partitions without multiplicity we have A325700, counted by A241638.
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. A000712, A001405, A026424, A028260, A239241 (conjugate A352129), A240009, A242498, A277579 (ranked by A349157).

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],Count[stc[#],?EvenQ]==Count[stc[#],?OddQ]&]

cp's OEIS Frontend

A338860 The excess of the number of partitions of n with more odd parts than even parts over the number of partitions of n with more even parts than odd parts.

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