

A276057


Sum of the asymmetry degrees of all compositions of n with parts in {1,3}.


2



0, 0, 0, 0, 2, 2, 4, 6, 14, 18, 36, 50, 94, 130, 236, 330, 580, 816, 1404, 1984, 3354, 4758, 7932, 11286, 18600, 26532, 43308, 61908, 100232, 143540, 230776, 331008, 528950, 759726, 1207584, 1736534, 2747242, 3954826, 6230444, 8977686, 14090410, 20320854
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OFFSET

0,5


COMMENTS

The asymmetry degree of a finite sequence of numbers is defined to be the number of pairs of symmetrically positioned distinct entries. Example: the asymmetry degree of (2,7,6,4,5,7,3) is 2, counting the pairs (2,3) and (6,5).
A sequence is palindromic if and only if its asymmetry degree is 0.


REFERENCES

S. Heubach and T. Mansour, Combinatorics of Compositions and Words, CRC Press, 2010.


LINKS



FORMULA

G.f.: g(z) = 2*z^4/((1+z+z^3)(1zz^3)^2). In the more general situation of compositions into a[1]<a[2]<a[3]<..., denoting F(z) = Sum(z^{a[j]},j>=1}, we have g(z) = (F(z)^2  F(z^2))/((1+F(z))(1F(z))^2).


EXAMPLE

a(6) = 4 because the compositions of 6 with parts in {1,3} are 33, 3111, 1311, 1131, 1113, and 111111 and the sum of their asymmetry degrees is 0 + 1+1+1+1+0.


MAPLE

g:=2*z^4/((1+z+z^3)*(1zz^3)^2): gser:=series(g, z=0, 45): seq(coeff(gser, z, n), n=0..40);


MATHEMATICA

Table[Total@ Map[Total, Map[Map[Boole[# >= 1] &, BitXor[Take[#  1, Ceiling[Length[#]/2]], Reverse@ Take[#  1, Ceiling[Length[#]/2]]]] &, Flatten[Map[Permutations, DeleteCases[IntegerPartitions@ n, {___, a_, ___} /; Nor[a == 1, a == 3]]], 1]]], {n, 0, 34}] // Flatten (* or *)
CoefficientList[Series[2 x^4/((1 + x + x^3) (1  x  x^3)^2), {x, 0, 41}], x] (* Michael De Vlieger, Aug 28 2016 *)


PROG

(PARI) concat(vector(4), Vec(2*x^4/((1+x+x^3)*(1xx^3)^2) + O(x^50))) \\ Colin Barker, Aug 28 2016


CROSSREFS



KEYWORD

nonn,easy


AUTHOR



STATUS

approved



