OFFSET
1,2
COMMENTS
Number of partitions of n such that all differences between successive parts are even, see example. [Joerg Arndt, Dec 27 2012]
Number of partitions of n where either all parts are odd or all parts are even. - Omar E. Pol, Aug 16 2013
From Gus Wiseman, Jan 13 2022: (Start)
Also the number of integer partitions of n with all even multiplicities (or run-lengths) except possibly the first. These are the conjugates of the partitions described by Joerg Arndt above. For example, the a(1) = 1 through a(8) = 11 partitions are:
(1) (2) (3) (4) (5) (6) (7) (8)
(11) (111) (22) (311) (33) (322) (44)
(211) (11111) (222) (511) (422)
(1111) (411) (31111) (611)
(2211) (1111111) (2222)
(21111) (3311)
(111111) (22211)
(41111)
(221111)
(2111111)
(11111111)
(End)
REFERENCES
A. G. Elashvili and V. G. Kac, Classification of good gradings of simple Lie algebras. Lie groups and invariant theory, 85-104, Amer. Math. Soc. Transl. Ser. 2, 213, Amer. Math. Soc., Providence, RI, 2005.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..1000
Karin Baur and Nolan Wallach, Nice parabolic subalgebras of reductive Lie algebras, Represent. Theory 9 (2005), 1-29.
A. G. Elashvili and V. G. Kac, Classification of good gradings of simple Lie algebras, arXiv:math-ph/0312030, 2002-2004.
FORMULA
G.f.: sum(j>=1, q^j * (1-q^j)/prod(i=1..j, 1-q^(2*i) ) ).
G.f.: F + G - 2, where F = prod(j>=1, 1/(1-q^(2*j) ), G = prod(j>=0, 1/(1-q^(2*j+1)) ).
EXAMPLE
From Joerg Arndt, Dec 27 2012: (Start)
There are a(10)=17 partitions of 10 where all differences between successive parts are even:
[ 1] [ 1 1 1 1 1 1 1 1 1 1 ]
[ 2] [ 2 2 2 2 2 ]
[ 3] [ 3 1 1 1 1 1 1 1 ]
[ 4] [ 3 3 1 1 1 1 ]
[ 5] [ 3 3 3 1 ]
[ 6] [ 4 2 2 2 ]
[ 7] [ 4 4 2 ]
[ 8] [ 5 1 1 1 1 1 ]
[ 9] [ 5 3 1 1 ]
[10] [ 5 5 ]
[11] [ 6 2 2 ]
[12] [ 6 4 ]
[13] [ 7 1 1 1 ]
[14] [ 7 3 ]
[15] [ 8 2 ]
[16] [ 9 1 ]
[17] [ 10 ]
(End)
MAPLE
b:= proc(n, i) option remember; `if`(i>n, 0,
`if`(irem(n, i)=0, 1, 0) +add(`if`(irem(j, 2)=0,
b(n-i*j, i+1), 0), j=0..n/i))
end:
a:= n-> b(n, 1):
seq(a(n), n=1..60); # Alois P. Heinz, Mar 26 2014
MATHEMATICA
(* The following Mathematica program first generates all of the palindromic, unimodal compositions of n and then counts them. *)
Pal[n_] := Block[{i, j, k, m, Q, L}, If[n == 1, Return[{{1}}]]; If[n == 2, Return[{{1, 1}, {2}}]]; L = {{n}}; If[Mod[n, 2] == 0, L = Append[L, {n/2, n/2}]]; For[i = 1, i < n, i++, Q = Pal[n - 2i]; m = Length[Q]; For[j = 1, j <= m, j++, If[i <= Q[[j, 1]], L = Append[L, Append[Prepend[Q[[j]], i], i]]]]]; L] NoPal[n_] := Length[Pal[n]]
a[n_] := PartitionsQ[n] + If[EvenQ[n], PartitionsP[n/2], 0]; Table[a[n], {n, 1, 55}] (* Jean-François Alcover, Mar 17 2014, after Vladeta Jovovic *)
Table[Length[Select[IntegerPartitions[n], And@@EvenQ/@Rest[Length/@Split[#]]&]], {n, 1, 30}] (* Gus Wiseman, Jan 13 2022 *)
PROG
(PARI) x='x+O('x^66); Vec(eta(x^2)/eta(x)+1/eta(x^2)-2) \\ Joerg Arndt, Jan 17 2016
CROSSREFS
The complement in partitions is counted by A006477
A version for compositions is A016116.
A000041 counts integer partitions.
A025065 counts palindromic partitions.
A027187 counts partitions with even length/maximum.
A035377 counts partitions using multiples of 3.
A340785 counts factorizations into even factors.
KEYWORD
nonn
AUTHOR
Nolan R. Wallach (nwallach(AT)ucsd.edu), Aug 10 2004
STATUS
approved