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A096441 Number of palindromic and unimodal compositions of n. Equivalently, the number of orbits under conjugation of even nilpotent n X n matrices. 7

%I

%S 1,2,2,4,3,7,5,11,8,17,12,26,18,37,27,54,38,76,54,106,76,145,104,199,

%T 142,266,192,357,256,472,340,621,448,809,585,1053,760,1354,982,1740,

%U 1260,2218,1610,2818,2048,3559,2590,4485,3264,5616,4097,7018,5120,8728,6378

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

%C Number of partitions of n such that all differences between successive parts are even, see example. [_Joerg Arndt_, Dec 27 2012]

%C Number of partitions of n where either all parts are odd or all parts are even. - _Omar E. Pol_, Aug 16 2013

%D 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.

%H Alois P. Heinz, <a href="/A096441/b096441.txt">Table of n, a(n) for n = 1..1000</a>

%H Karin Baur and Nolan Wallach, <a href="http://dx.doi.org/10.1090/S1088-4165-05-00262-1">Nice parabolic subalgebras of reductive Lie algebras</a>, Represent. Theory 9 (2005), 1-29.

%H A. G. Elashvili and V. G. Kac, <a href="http://arxiv.org/abs/math-ph/0312030">Classification of good gradings of simple Lie algebras</a>, arXiv:math-ph/0312030, 2002-2004.

%F G.f.: sum(j>=1, q^j * (1-q^j)/prod(i=1..j, 1-q^(2*i) ) ).

%F 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)) ).

%F a(2*n) = A000041(n) + A000009(2*n); a(2*n-1) = A000009(2*n-1). - _Vladeta Jovovic_, Aug 11 2004

%F a(n) = A000009(n) + A035363(n) = A000041(n) - A006477(n). - _Omar E. Pol_, Aug 16 2013

%e From _Joerg Arndt_, Dec 27 2012: (Start)

%e There are a(10)=17 partitions of 10 where all differences between successive parts are even:

%e [ 1] [ 1 1 1 1 1 1 1 1 1 1 ]

%e [ 2] [ 2 2 2 2 2 ]

%e [ 3] [ 3 1 1 1 1 1 1 1 ]

%e [ 4] [ 3 3 1 1 1 1 ]

%e [ 5] [ 3 3 3 1 ]

%e [ 6] [ 4 2 2 2 ]

%e [ 7] [ 4 4 2 ]

%e [ 8] [ 5 1 1 1 1 1 ]

%e [ 9] [ 5 3 1 1 ]

%e [10] [ 5 5 ]

%e [11] [ 6 2 2 ]

%e [12] [ 6 4 ]

%e [13] [ 7 1 1 1 ]

%e [14] [ 7 3 ]

%e [15] [ 8 2 ]

%e [16] [ 9 1 ]

%e [17] [ 10 ]

%e (End)

%p b:= proc(n, i) option remember; `if`(i>n, 0,

%p `if`(irem(n, i)=0, 1, 0) +add(`if`(irem(j, 2)=0,

%p b(n-i*j, i+1), 0), j=0..n/i))

%p end:

%p a:= n-> b(n, 1):

%p seq(a(n), n=1..60); # _Alois P. Heinz_, Mar 26 2014

%t (* The following Mathematica program first generates all of the palindromic, unimodal compositions of n and then counts them. *)

%t 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]]

%t a[n_] := PartitionsQ[n] + If[EvenQ[n], PartitionsP[n/2], 0]; Table[a[n], {n, 1, 55}] (* _Jean-Fran├žois Alcover_, Mar 17 2014, after _Vladeta Jovovic_ *)

%o (PARI) x='x+O('x^66); Vec(eta(x^2)/eta(x)+1/eta(x^2)-2) \\ _Joerg Arndt_, Jan 17 2016

%Y Bisections are A078408 and A096967.

%K nonn

%O 1,2

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

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Last modified June 4 08:18 EDT 2020. Contains 334825 sequences. (Running on oeis4.)