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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 (list; graph; refs; listen; history; text; internal format)
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
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.
FORMULA
G.f.: g(z) = 2*z^4/((1+z+z^3)(1-z-z^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))(1-F(z))^2).
a(n) = Sum_{k>=0} k*A276056(n,k).
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)*(1-z-z^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)*(1-x-x^3)^2) + O(x^50))) \\ Colin Barker, Aug 28 2016
CROSSREFS
Cf. A276056.
Sequence in context: A181926 A061894 A116684 * A116637 A153961 A134041
KEYWORD
nonn,easy
AUTHOR
Emeric Deutsch, Aug 18 2016
STATUS
approved

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Last modified June 18 00:38 EDT 2024. Contains 373468 sequences. (Running on oeis4.)