A096441 Number of palindromic and unimodal compositions of n. Equivalently, the number of orbits under conjugation of even nilpotent n X n matrices.

1, 2, 2, 4, 3, 7, 5, 11, 8, 17, 12, 26, 18, 37, 27, 54, 38, 76, 54, 106, 76, 145, 104, 199, 142, 266, 192, 357, 256, 472, 340, 621, 448, 809, 585, 1053, 760, 1354, 982, 1740, 1260, 2218, 1610, 2818, 2048, 3559, 2590, 4485, 3264, 5616, 4097, 7018, 5120, 8728, 6378

Offset: 1

Author

Nolan R. Wallach (nwallach(AT)ucsd.edu), Aug 10 2004

Keywords

• nonn

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)

Examples

			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)
		

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.
• Sergi Elizalde and Emeric Deutsch, The degree of asymmetry of a sequence, Enum. Combinat. Applic. 2 (2022) no 1 #S2R7, U(0,z).

Crossrefs

Bisections are A078408 and A096967.

The complement in partitions is counted by A006477

A version for compositions is A016116.

A pointed version is A035363, ranked by A066207.

A000041 counts integer partitions.

A025065 counts palindromic partitions.

A027187 counts partitions with even length/maximum.

A035377 counts partitions using multiples of 3.

A058696 counts partitions of even numbers, ranked by A300061.

A340785 counts factorizations into even factors.

Cf. A000009, A002865, A027383, A035457, A117298, A117989, A168021, A274230, A345170, A349060, A349061.

Programs

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 *)

PARI

my(x='x+O('x^66)); Vec(eta(x^2)/eta(x)+1/eta(x^2)-2) \\ Joerg Arndt, Jan 17 2016

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 = Product_{j>=1} 1/(1-q^(2*j)), G = Product_{j>=0} 1/(1-q^(2*j+1)).

a(2*n) = A000041(n) + A000009(2*n); a(2*n-1) = A000009(2*n-1). - Vladeta Jovovic, Aug 11 2004

a(n) = A000009(n) + A035363(n) = A000041(n) - A006477(n). - Omar E. Pol, Aug 16 2013