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A276060 Triangle read by rows: T(n,k) is the number of compositions of n into parts congruent to 1 mod 3 and having asymmetry degree equal to k (n>=0; 0<=k<=floor(n/5)).

%I A276060 #16 Apr 12 2025 03:40:55
%S A276060 1,1,1,1,2,1,2,2,2,2,4,3,6,3,10,5,10,4,4,20,4,7,22,12,6,34,20,10,42,
%T A276060 36,9,64,48,8,15,70,96,8,13,112,120,32,22,124,204,56,19,184,280,112,
%U A276060 32,212,436,176,16,28,310,564,360,16,47,346,896,504,80,41,512,1128,920,144,69,570,1704,1360
%N A276060 Triangle read by rows: T(n,k) is the number of compositions of n into parts congruent to 1 mod 3 and having asymmetry degree equal to k (n>=0; 0<=k<=floor(n/5)).
%C A276060 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 A276060 Number of entries in row n is 1 + floor(n/5).
%C A276060 Sum of entries in row n is A000930(n-1).
%C A276060 T(n,0) = A226916(n+6) (n>=1).
%C A276060 Sum_{k>=0} k*T(n,k) = A276061(n).
%D A276060 S. Heubach and T. Mansour, Combinatorics of Compositions and Words, CRC Press, 2010.
%H A276060 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 A276060 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.
%F A276060 G.f.: G(t,z) = (1-z^2)*(1+z+z^2)*(1-z+z^2)*(1+z-z^3)/(1-z^2-z^3+z^5-2*t*z^5-z^6+z^9). In the more general situation of compositions into a[1]<a[2]<a[3]<..., denoting F(z) = Sum_{j>=1} z^(a[j]), we have G(t,z) = (1 + F(z))/(1 - F(z^2) - t*(F(z)^2 - F(z^2))). In particular, for t=0 we obtain Theorem 1.2 of the Hoggatt et al. reference.
%e A276060 Row 7 is [2,4] because the compositions of 7 with parts in {1,4,7,10,...} are 7, 4111, 1411, 1141, 1114, and 1111111, having asymmetry degrees 0, 1, 1, 1, 1, and 0, respectively.
%e A276060 Triangle starts:
%e A276060   1;
%e A276060   1;
%e A276060   1;
%e A276060   1;
%e A276060   2;
%e A276060   1,2;
%e A276060   2,2.
%p A276060 G := (1-z^2)*(1+z+z^2)*(1-z+z^2)*(1+z-z^3)/(1-z^2-z^3+z^5-2*t*z^5-z^6+z^9): Gser := simplify(series(G, z = 0, 30)): for n from 0 to 25 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 25 do seq(coeff(P[n], t, j), j = 0 .. degree(P[n])) end do; # yields sequence in triangular form
%t A276060 Table[TakeWhile[BinCounts[#, {0, 1 + Floor[n/5], 1}], # != 0 &] &@ 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, 24}] // Flatten (* _Michael De Vlieger_, Aug 22 2016 *)
%Y A276060 Cf. A000930, A226916, A276061.
%K A276060 nonn,tabf
%O A276060 0,5
%A A276060 _Emeric Deutsch_, Aug 22 2016