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A096441
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Number of palindromic and unimodal compositions of n. Equivalently, the number of orbits under conjugation of even nilpotent n X n matrices.
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29
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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
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OFFSET
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1,2
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COMMENTS
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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
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)
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REFERENCES
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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.
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LINKS
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FORMULA
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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)) ).
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EXAMPLE
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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)
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MAPLE
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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):
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MATHEMATICA
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(* 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]]
Table[Length[Select[IntegerPartitions[n], And@@EvenQ/@Rest[Length/@Split[#]]&]], {n, 1, 30}] (* Gus Wiseman, Jan 13 2022 *)
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PROG
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(PARI) x='x+O('x^66); Vec(eta(x^2)/eta(x)+1/eta(x^2)-2) \\ Joerg Arndt, Jan 17 2016
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CROSSREFS
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The complement in partitions is counted by A006477
A version for compositions is A016116.
A025065 counts palindromic partitions.
A027187 counts partitions with even length/maximum.
A035377 counts partitions using multiples of 3.
A340785 counts factorizations into even factors.
Cf. A000009, A002865, A027383, A035457, A117298, A117989, A168021, A274230, A345170, A349060, A349061.
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KEYWORD
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nonn
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AUTHOR
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Nolan R. Wallach (nwallach(AT)ucsd.edu), Aug 10 2004
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STATUS
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approved
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