login
Sum of the asymmetry degrees of all compositions of n into parts congruent to 1 mod 3.
2

%I #13 Aug 28 2016 11:09:16

%S 0,0,0,0,0,2,2,4,6,10,18,28,46,74,114,184,286,448,700,1080,1676,2582,

%T 3970,6104,9338,14288,21808,33224,50580,76844,116640,176832,267740,

%U 405058,612110,924204,1394266,2101558,3165406,4764184,7165530,10770386,16178378

%N Sum of the asymmetry degrees of all compositions of n into parts congruent to 1 mod 3.

%C 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).

%C A sequence is palindromic if and only if its asymmetry degree is 0.

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

%H Colin Barker, <a href="/A276061/b276061.txt">Table of n, a(n) for n = 0..1000</a>

%H Krithnaswami Alladi and V. E. Hoggatt, Jr. <a href="http://www.fq.math.ca/Scanned/13-3/alladi1.pdf">Compositions with Ones and Twos</a>, Fibonacci Quarterly, 13 (1975), 233-239.

%H V. E. Hoggatt, Jr., and Marjorie Bicknell, <a href="http://www.fq.math.ca/Scanned/13-4/hoggatt1.pdf">Palindromic compositions</a>, Fibonacci Quart., Vol. 13(4), 1975, pp. 350-356.

%H <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,1,-1,0,-1,-1,-1,-2,1,0,1).

%F G.f. g(z) = 2*z^5*(1-z^3)/((1+z)(1-z+z^2)(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).

%F a(n) = Sum(k*A276060(n,k), k>=0).

%e a(7) = 4 because the compositions of 7 with parts in {1,4,7,10,... } are 7, 4111, 1411, 1141, 1114, and 1111111, and the sum of their asymmetry degrees is 0+1+1+1+1+0.

%p g := 2*z^5*(1-z^3)/((1+z)*(1-z+z^2)*(1+z-z^3)*(1-z-z^3)^2): gser := series(g, z = 0, 45): seq(coeff(gser, z, n), n = 0 .. 40);

%t 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_, ___} /; Mod[a, 3] != 1]], 1]]], {n, 0, 36}] // Flatten (* _Michael De Vlieger_, Aug 22 2016 *)

%o (PARI) concat(vector(5), Vec(2*x^5*(1-x^3)/((1+x)*(1-x+x^2)*(1+x-x^3)*(1-x-x^3)^2) + O(x^50))) \\ _Colin Barker_, Aug 28 2016

%Y Cf. A276060.

%K nonn,easy

%O 0,6

%A _Emeric Deutsch_, Aug 22 2016