|
%I #36 Jan 16 2022 11:08:51
%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
%C From _Gus Wiseman_, Jan 13 2022: (Start)
%C 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:
%C (1) (2) (3) (4) (5) (6) (7) (8)
%C (11) (111) (22) (311) (33) (322) (44)
%C (211) (11111) (222) (511) (422)
%C (1111) (411) (31111) (611)
%C (2211) (1111111) (2222)
%C (21111) (3311)
%C (111111) (22211)
%C (41111)
%C (221111)
%C (2111111)
%C (11111111)
%C (End)
%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-François Alcover_, Mar 17 2014, after _Vladeta Jovovic_ *)
%t Table[Length[Select[IntegerPartitions[n],And@@EvenQ/@Rest[Length/@Split[#]]&]],{n,1,30}] (* _Gus Wiseman_, Jan 13 2022 *)
%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.
%Y The complement in partitions is counted by A006477
%Y A version for compositions is A016116.
%Y A pointed version is A035363, ranked by A066207.
%Y A000041 counts integer partitions.
%Y A025065 counts palindromic partitions.
%Y A027187 counts partitions with even length/maximum.
%Y A035377 counts partitions using multiples of 3.
%Y A058696 counts partitions of even numbers, ranked by A300061.
%Y A340785 counts factorizations into even factors.
%Y Cf. A000009, A002865, A027383, A035457, A117298, A117989, A168021, A274230, A345170, A349060, A349061.
%K nonn
%O 1,2
%A Nolan R. Wallach (nwallach(AT)ucsd.edu), Aug 10 2004
|
