OFFSET
0,7
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
Colin Barker, Table of n, a(n) for n = 0..1000
Krithnaswami Alladi and V. E. Hoggatt, Jr. Compositions with Ones and Twos, Fibonacci Quarterly, 13 (1975), 233-239.
V. E. Hoggatt, Jr., and Marjorie Bicknell, Palindromic compositions, Fibonacci Quart., Vol. 13(4), 1975, pp. 350-356.
Index entries for linear recurrences with constant coefficients, signature (1,1,-1,0,1,2,-3,0,0,1,-3,0,0,0,-1).
FORMULA
G.f.: g(z) = 2*z^6/((1-z+z^2)^2*(1-z^2-z^3)^2*(1+z+z^2)(1-z^2+z^3)). 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))(1-F(z))^2).
a(n) = Sum(k*A276064(n,k), k>=0).
EXAMPLE
a(8) = 4 because the compositions of 8 with parts in {1,5} are 5111, 1511, 1151, 1115, and 11111111, and the sum of their asymmetry degrees is 1+1+1+1+0.
MAPLE
g := 2*z^6/((1-z+z^2)^2*(1-z^2-z^3)^2*(1+z+z^2)*(1-z^2+z^3)): 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 == 5]]], 1]]], {n, 0, 42}] // Flatten (* Michael De Vlieger, Aug 22 2016 *)
LinearRecurrence[{1, 1, -1, 0, 1, 2, -3, 0, 0, 1, -3, 0, 0, 0, -1}, {0, 0, 0, 0, 0, 0, 2, 2, 4, 4, 6, 8, 16, 20, 34}, 50] (* Harvey P. Dale, Aug 29 2021 *)
PROG
(PARI) concat(vector(6), Vec(2*x^6/((1-x+x^2)^2*(1-x^2-x^3)^2*(1+x+x^2)*(1-x^2+x^3)) + O(x^50))) \\ Colin Barker, Aug 28 2016
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Emeric Deutsch, Aug 22 2016
STATUS
approved