A236913 - OEIS

%I #23 Feb 11 2021 22:59:47

%S 1,1,3,6,12,22,40,69,118,195,317,505,793,1224,1867,2811,4186,6168,

%T 9005,13026,18692,26613,37619,52815,73680,102162,140853,193144,263490,

%U 357699,483338,650196,870953,1161916,1544048,2044188,2696627,3545015,4644850,6066425

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

%C The partitions of n are partitioned into four types:

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

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

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

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

%C A236559 and A160786 are the bisections of A027193;

%C A236913 and A236914 are the bisections of A027187.

%H Alois P. Heinz, <a href="/A236913/b236913.txt">Table of n, a(n) for n = 0..1000</a>

%e The partitions of 4 of type EE are [3,1], [2,2], [1,1,1,1], so that a(2) = 3.

%e type/k . 1 .. 2 .. 3 .. 4 .. 5 .. 6 .. 7 .. 8 ... 9 ... 10 .. 11

%e EO ..... 0 .. 1 .. 0 .. 2 .. 0 .. 5 .. 0 .. 10 .. 0 ... 20 .. 0

%e OE ..... 1 .. 0 .. 2 .. 0 .. 4 .. 0 .. 8 .. 0 ... 16 .. 0 ... 29

%e EE ..... 0 .. 1 .. 0 .. 3 .. 0 .. 6 .. 0 .. 12 .. 0 ... 22 .. 0

%e OO ..... 0 .. 0 .. 1 .. 0 .. 3 .. 0 .. 7 .. 0 ... 14 .. 0 ... 27

%e From _Gus Wiseman_, Feb 09 2021: (Start)

%e 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:

%e   ()  (11)  (22)    (33)      (44)

%e             (31)    (42)      (53)

%e             (1111)  (51)      (62)

%e                     (2211)    (71)

%e                     (3111)    (2222)

%e                     (111111)  (3221)

%e                               (3311)

%e                               (4211)

%e                               (5111)

%e                               (221111)

%e                               (311111)

%e                               (11111111)

%e (End)

%p b:= proc(n, i) option remember; `if`(n=0, [1, 0$3],

%p       `if`(i<1, [0$4], b(n, i-1)+`if`(i>n, [0$4], (p->

%p       `if`(irem(i, 2)=0, [p[3], p[4], p[1], p[2]],

%p           [p[2], p[1], p[4], p[3]]))(b(n-i, i)))))

%p     end:

%p a:= n-> b(2*n$2)[1]:

%p seq(a(n), n=0..40);  # _Alois P. Heinz_, Feb 16 2014

%t 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]} &,

%t OddQ[IntegerPartitions[2 #]]], EvenQ[(*Odd*)First[#]] && EvenQ[(*Even*)Last[#]] &]] &, Range[z]] ; m4 = Map[Length[Select[Map[{Count[#, True], Count[#, False]} &,

%t OddQ[IntegerPartitions[2 # - 1]]], OddQ[(*Odd*)First[#]] && OddQ[(*Even*)Last[#]] &]] &, Range[z]];

%t m1 (* A236559, type EO*)

%t m2 (* A160786, type OE*)

%t m3 (* A236913, type EE*)

%t m4 (* A236914, type OO*)

%t (* _Peter J. C. Moses_, Feb 03 2014 *)

%t 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_ *)

%t Table[Length[Select[IntegerPartitions[2n],EvenQ[Length[#]]&]],{n,0,15}] (* _Gus Wiseman_, Feb 09 2021 *)

%Y Cf. A000041, A027193, A236559, A236914.

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

%Y The ordered version is A000302.

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

%Y The Heinz numbers of these partitions are (A340784).

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

%Y A034008 counts compositions of even length.

%Y A035363 counts partitions into even parts (A066207).

%Y A047993 counts balanced partitions (A106529).

%Y A058695 counts partitions of odd numbers (A300063).

%Y A058696 counts partitions of even numbers (A300061).

%Y A067661 counts strict partitions of even length (A030229).

%Y A072233 counts partitions by sum and length.

%Y A339846 counts factorizations of even length.

%Y A340601 counts partitions of even rank (A340602).

%Y A340785 counts factorizations into even factors.

%Y A340786 counts even-length factorizations into even factors.

%K nonn,easy

%O 0,3

%A _Clark Kimberling_, Feb 01 2014

%E More terms from _Alois P. Heinz_, Feb 16 2014