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A276056 Triangle read by rows: T(n,k) is the number of compositions of n with parts in {1,3} and having asymmetry degree equal to k, (n>=0; 0<=k<=floor(n/4)).

%I A276056 #23 Apr 12 2025 03:41:10
%S A276056 1,1,1,2,1,2,2,2,2,4,3,6,3,6,4,5,10,4,4,12,12,7,18,16,6,22,24,8,10,34,
%T A276056 36,8,9,36,52,32,15,58,76,40,13,60,108,80,16,22,96,160,112,16,19,100,
%U A276056 204,192,80,32,160,312,272,96,28,162,376,440,240,32
%N A276056 Triangle read by rows: T(n,k) is the number of compositions of n with parts in {1,3} and having asymmetry degree equal to k, (n>=0; 0<=k<=floor(n/4)).
%C A276056 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 A276056 Number of entries in row n is 1 + floor(n/4).
%C A276056 Sum of entries in row n is A000930(n).
%D A276056 S. Heubach and T. Mansour, Combinatorics of Compositions and Words, CRC Press, 2010.
%H A276056 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 A276056 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 A276056 G.f.: G(t,z) = (1+z+z^3)/(1-z^2-2*t*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.
%F A276056 T(n,0) = A226916(n+7).
%F A276056 Sum_{k>=0} k*T(n,k) = A276057(n).
%e A276056 Row 6 is [2,4] because the compositions of 6 with parts in {1,3} are 33, 3111, 1311, 1131, 1113, and 111111, having asymmetry degrees 0, 1, 1, 1, 1, and 0, respectively.
%e A276056 Triangle starts:
%e A276056   1;
%e A276056   1;
%e A276056   1;
%e A276056   2;
%e A276056   1, 2;
%e A276056   2, 2;
%e A276056   2, 4;
%e A276056   ...
%p A276056 G := (1+z+z^3)/(1-z^2-2*t*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 A276056 Table[TakeWhile[BinCounts[#, {0, 1 + Floor[n/4], 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_, ___} /; Nor[a == 1, a == 3]]], 1]]], {n, 0, 20}] // Flatten (* _Michael De Vlieger_, Aug 28 2016 *)
%Y A276056 Cf. A000930, A226916, A276057.
%K A276056 nonn,tabf
%O A276056 0,4
%A A276056 _Emeric Deutsch_, Aug 18 2016