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A275446 Triangle read by rows: T(n,k) is the number of compositions of n with parts in {2,1,3,5,7,9,...} and having asymmetry degree equal to k (n>=0; 0<=k<=floor(n/3)).

%I A275446 #17 Apr 12 2025 01:01:10
%S A275446 1,1,2,2,2,3,4,4,10,6,16,4,8,30,12,11,48,36,15,82,76,8,21,128,164,32,
%T A275446 29,204,312,112,40,312,596,288,16,55,482,1064,704,80,76,728,1884,1536,
%U A275446 320,105,1100,3212,3248,960,32,145,1640,5428,6464,2624,192
%N A275446 Triangle read by rows: T(n,k) is the number of compositions of n with parts in {2,1,3,5,7,9,...} and having asymmetry degree equal to k (n>=0; 0<=k<=floor(n/3)).
%C A275446 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 A275446 number of entries in row n is 1+floor(n/3).
%C A275446 Sum of entries in row n is A052535(n).
%C A275446 T(n,0) = A103632(n+2).
%C A275446 Sum_{k>=0} k*T(n,k) = A275447(n).
%D A275446 S. Heubach and T. Mansour, Combinatorics of Compositions and Words, CRC Press, 2010.
%H A275446 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 A275446 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 A275446 G.f.: G(t,z) = (1-z^4)/(1-z-z^2+(1-2*t)*z^3-z^4+z^6). 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 A275446 Row 4 is [3,4] because the compositions of 4 with parts in {2,1,3,5,7,...} are 22, 31, 13, 211, 121, 112, and 1111, having asymmetry degrees 0, 1, 1, 1, 0, 1, and 0, respectively.
%e A275446 Triangle starts:
%e A275446   1;
%e A275446   1;
%e A275446   2;
%e A275446   2,2;
%e A275446   3,4;
%e A275446   4,10;
%e A275446   6,16,4.
%p A275446 G := (1-z^4)/(1-z-z^2+(1-2*t)*z^3-z^4+z^6): 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 A275446 Table[BinCounts[#, {0, 1 + Floor[n/3], 1}] &@ 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_, ___} /; And[EvenQ@ a, a != 2]]], 1]]], {n, 0, 16}] // Flatten (* _Michael De Vlieger_, Aug 17 2016 *)
%Y A275446 Cf. A052535, A103632, A275447.
%K A275446 nonn,tabf
%O A275446 0,3
%A A275446 _Emeric Deutsch_, Aug 17 2016