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A236913
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Number of partitions of 2n of type EE (see Comments).
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49
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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
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OFFSET
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0,3
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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
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)
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MAPLE
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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]:
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MATHEMATICA
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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]];
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 *)
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CROSSREFS
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Note: A-numbers of ranking sequences are in parentheses below.
The case of odd-length partitions of odd numbers is A160786 (A340931).
The Heinz numbers of these partitions are (A340784).
A034008 counts compositions of even length.
A072233 counts partitions by sum and length.
A339846 counts factorizations of even length.
A340785 counts factorizations into even factors.
A340786 counts even-length factorizations into even factors.
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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