A108949 Number of partitions of n with more even parts than odd parts.

0, 0, 1, 0, 2, 1, 3, 3, 6, 7, 10, 14, 19, 26, 33, 45, 58, 77, 97, 127, 161, 205, 259, 326, 411, 510, 639, 786, 980, 1197, 1482, 1800, 2216, 2677, 3275, 3942, 4793, 5749, 6951, 8309, 9995, 11912, 14259, 16944, 20194, 23926, 28402, 33559, 39687, 46767, 55120, 64780, 76110, 89222

Offset: 0

Author

Len Smiley, Jul 21 2005

Keywords

• nonn

Examples

			a(6) = 3: {[6], [4,2], [2,2,2]}; a(7) = 3: {[4,2,1], [3,2,2], [2,2,2,1]}.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• B. Kim, E. Kim, and J. Lovejoy, Parity bias in partitions, European J. Combin., 89 (2020), 103159, 19 pp.

Crossrefs

Cf. A045931 for #even parts = #odd parts, A108950 for #even parts < #odd parts.

Cf. A171966, A130780. - Reinhard Zumkeller, Jan 21 2010

Programs

Maple

with(combinat,partition):
evnbigrodd:=proc(n::nonnegint)
   local evencount,oddcount,bigcount,parts,i,j;
   bigcount:=0;
   partitions:=partition(n);
   for i from 1 to nops(partitions) do
      evencount:=0;
      oddcount:=0;
      for j from 1 to nops(partitions[i]) do
         if (op(j,partitions[i]) mod 2 <>0) then
            oddcount:=oddcount+1
         fi;
         if (op(j,partitions[i]) mod 2 =0) then
            evencount:=evencount+1
         fi
      od;
      if (evencount>oddcount) then
         bigcount:=bigcount+1
      fi
   od;
   return(bigcount)
end proc;
seq(evnbigrodd(i),i=1..42);
# second Maple program:
b:= proc(n, i, t) option remember; `if`(n=0,
      `if`(t<0, 1, 0), `if`(i<1, 0, b(n, i-1, t)+
      `if`(i>n, 0, b(n-i, i, t+(2*irem(i, 2)-1)))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..80);  # Alois P. Heinz, Mar 30 2014

Mathematica

p[n_] := p[n] = Select[IntegerPartitions[n], Count[#, ?OddQ] == Count[#, ?EvenQ] &]; t = Table[p[n], {n, 0, 10}] (* partitions of n with # odd parts = # even parts *)
TableForm[t] (* partitions, vertical format *)
Table[Length[p[n]], {n, 0, 30}] (* A045931 *)
(* Peter J. C. Moses, Mar 10 2014 *)
b[n_, i_, t_] := b[n, i, t] = If[n==0, If[t<0, 1, 0], If[i<1, 0, b[n, i-1, t] + If[i>n, 0, b[n-i, i, t+(2*Mod[i, 2]-1)]]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Nov 02 2015, after Alois P. Heinz *)

PARI

a(n) = {nb = 0; forpart(p=n, nb += (2*#(select(x->x%2, Vec(p))) < #p);); nb;} \\ Michel Marcus, Nov 02 2015

Formula

a(n) = A171966(n) - A045931(n) = A171967(n) - A108950(n). - Reinhard Zumkeller, Jan 21 2010

a(n) = Sum_{k=-floor(n/2)+(n mod 2)..-1} A240009(n,k). - Alois P. Heinz, Mar 30 2014

G.f.: (Product_{k>=1} 1/(1-x^(2*k-1)))*Sum_{n>=1} q^(2*n^2)*(1-q^(n))/Product_{k=1..n} (1-q^(2*k))^2. - Jeremy Lovejoy, Jan 12 2021

Extensions

More terms from Joerg Arndt, Oct 04 2012