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Sum of the asymmetry degrees of all compositions of n with parts in {1,3}.
2

%I #14 Jan 05 2025 19:51:41

%S 0,0,0,0,2,2,4,6,14,18,36,50,94,130,236,330,580,816,1404,1984,3354,

%T 4758,7932,11286,18600,26532,43308,61908,100232,143540,230776,331008,

%U 528950,759726,1207584,1736534,2747242,3954826,6230444,8977686,14090410,20320854

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

%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="/A276057/b276057.txt">Table of n, a(n) for n = 0..1000</a>

%H Krithnaswami Alladi and V. E. Hoggatt, Jr. <a href="https://web.archive.org/web/2024*/https://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="https://web.archive.org/web/2024*/https://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_09">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,0,2,-3,1,-3,0,-1).

%F 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).

%F a(n) = Sum_{k>=0} k*A276056(n,k).

%e 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.

%p 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);

%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_, ___} /; Nor[a == 1, a == 3]]], 1]]], {n, 0, 34}] // Flatten (* or *)

%t CoefficientList[Series[2 x^4/((1 + x + x^3) (1 - x - x^3)^2), {x, 0, 41}], x] (* _Michael De Vlieger_, Aug 28 2016 *)

%o (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

%Y Cf. A276056.

%K nonn,easy

%O 0,5

%A _Emeric Deutsch_, Aug 18 2016