A236913 Number of partitions of 2n of type EE (see Comments).

1, 1, 3, 6, 12, 22, 40, 69, 118, 195, 317, 505, 793, 1224, 1867, 2811, 4186, 6168, 9005, 13026, 18692, 26613, 37619, 52815, 73680, 102162, 140853, 193144, 263490, 357699, 483338, 650196, 870953, 1161916, 1544048, 2044188, 2696627, 3545015, 4644850, 6066425

Offset: 0

Author

Clark Kimberling, Feb 01 2014

Keywords

• nonn
• easy

Comments

The partitions of n are partitioned into four types:

EO, even # of odd parts and odd # of even parts, A236559;

OE, odd # of odd parts and even # of even parts, A160786;

EE, even # of odd parts and even # of even parts, A236913;

OO, odd # of odd parts and odd # of even parts, A236914.

A236559 and A160786 are the bisections of A027193;

A236913 and A236914 are the bisections of A027187.

Examples

			The partitions of 4 of type EE are [3,1], [2,2], [1,1,1,1], so that a(2) = 3.
type/k . 1 .. 2 .. 3 .. 4 .. 5 .. 6 .. 7 .. 8 ... 9 ... 10 .. 11
EO ..... 0 .. 1 .. 0 .. 2 .. 0 .. 5 .. 0 .. 10 .. 0 ... 20 .. 0
OE ..... 1 .. 0 .. 2 .. 0 .. 4 .. 0 .. 8 .. 0 ... 16 .. 0 ... 29
EE ..... 0 .. 1 .. 0 .. 3 .. 0 .. 6 .. 0 .. 12 .. 0 ... 22 .. 0
OO ..... 0 .. 0 .. 1 .. 0 .. 3 .. 0 .. 7 .. 0 ... 14 .. 0 ... 27
From _Gus Wiseman_, Feb 09 2021: (Start)
This sequence counts even-length partitions of even numbers, which have Heinz numbers given by A340784. For example, the a(0) = 1 through a(4) = 12 partitions are:
  ()  (11)  (22)    (33)      (44)
            (31)    (42)      (53)
            (1111)  (51)      (62)
                    (2211)    (71)
                    (3111)    (2222)
                    (111111)  (3221)
                              (3311)
                              (4211)
                              (5111)
                              (221111)
                              (311111)
                              (11111111)
(End)
		

Links

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

Crossrefs

Cf. A000041, A027193, A236559, A236914.

Note: A-numbers of ranking sequences are in parentheses below.

The ordered version is A000302.

The case of odd-length partitions of odd numbers is A160786 (A340931).

The Heinz numbers of these partitions are (A340784).

A027187 counts partitions of even length/maximum (A028260/A244990).

A034008 counts compositions of even length.

A035363 counts partitions into even parts (A066207).

A047993 counts balanced partitions (A106529).

A058695 counts partitions of odd numbers (A300063).

A058696 counts partitions of even numbers (A300061).

A067661 counts strict partitions of even length (A030229).

A072233 counts partitions by sum and length.

A339846 counts factorizations of even length.

A340601 counts partitions of even rank (A340602).

A340785 counts factorizations into even factors.

A340786 counts even-length factorizations into even factors.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, [1, 0$3],
      `if`(i<1, [0$4], b(n, i-1)+`if`(i>n, [0$4], (p->
      `if`(irem(i, 2)=0, [p[3], p[4], p[1], p[2]],
          [p[2], p[1], p[4], p[3]]))(b(n-i, i)))))
    end:
a:= n-> b(2*n$2)[1]:
seq(a(n), n=0..40);  # Alois P. Heinz, Feb 16 2014

Mathematica

z = 25; m1 = Map[Length[Select[Map[{Count[#, True], Count[#, False]} &,  OddQ[IntegerPartitions[2 #]]], EvenQ[(*Odd*)First[#]] && OddQ[(*Even*)Last[#]] &]] &, Range[z]]; m2 = Map[Length[Select[Map[{Count[#, True], Count[#, False]} &,      OddQ[IntegerPartitions[2 # - 1]]], OddQ[(*Odd*)First[#]] && EvenQ[(*Even*)Last[#]] &]] &, Range[z]]; m3 = Map[Length[Select[Map[{Count[#, True], Count[#, False]} &,
OddQ[IntegerPartitions[2 #]]], EvenQ[(*Odd*)First[#]] && EvenQ[(*Even*)Last[#]] &]] &, Range[z]] ; m4 = Map[Length[Select[Map[{Count[#, True], Count[#, False]} &,
OddQ[IntegerPartitions[2 # - 1]]], OddQ[(*Odd*)First[#]] && OddQ[(*Even*)Last[#]] &]] &, Range[z]];
m1 (* A236559, type EO*)
m2 (* A160786, type OE*)
m3 (* A236913, type EE*)
m4 (* A236914, type OO*)
(* Peter J. C. Moses, Feb 03 2014 *)
b[n_, i_] := b[n, i] = If[n == 0, {1, 0, 0, 0}, If[i < 1, {0, 0, 0, 0}, b[n, i - 1] + If[i > n, {0, 0, 0, 0}, Function[p, If[Mod[i, 2] == 0, p[[{3, 4, 1, 2}]], p[[{2, 1, 4, 3}]]]][b[n - i, i]]]]]; a[n_] := b[2*n, 2*n][[1]]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Oct 27 2015, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[2n],EvenQ[Length[#]]&]],{n,0,15}] (* Gus Wiseman, Feb 09 2021 *)

Extensions

More terms from Alois P. Heinz, Feb 16 2014