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A276058 Triangle read by rows: T(n,k) is the number of compositions of n with parts in {3,4,5,6,...} and having asymmetry degree equal to k (n>=0; 0<=k<=floor(n/7)).

%I A276058 #13 Apr 12 2025 03:41:03
%S A276058 1,1,1,1,2,1,2,2,2,2,4,3,6,3,10,5,14,4,24,7,30,4,6,46,8,10,58,20,9,84,
%T A276058 36,15,106,68,13,152,112,22,188,196,19,264,304,8,32,324,492,24,28,446,
%U A276058 732,72,47,546,1120,160,41,744,1616,344
%N A276058 Triangle read by rows: T(n,k) is the number of compositions of n with parts in {3,4,5,6,...} and having asymmetry degree equal to k (n>=0; 0<=k<=floor(n/7)).
%C A276058 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 A276058 A sequence is palindromic if and only if its asymmetry degree i 0.
%C A276058 Number of entries in row n is 1 + floor(n/7).
%C A276058 Sum of entries in row n is A078012(n).
%C A276058 T(n,0) = A226916(n+4)
%C A276058 Sum_{k>=0} k*T(n,k) = A276059(n).
%D A276058 S. Heubach and T. Mansour, Combinatorics of Compositions and Words, CRC Press, 2010.
%H A276058 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 A276058 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 A276058 G.f.: G(t,z) = (1-z^2)*(1-z+z^3)/(1-z-z^2+z^3-z^6+z^7-2*t*z^7). 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 A276058 Row 7 is [1,2] because the compositions of 7 with parts in {3,4,5,...} are 7, 34, and 43, having asymmetry degrees 0, 1, and 1, respectively.
%e A276058 Triangle starts:
%e A276058   1;
%e A276058   0;
%e A276058   0;
%e A276058   1;
%e A276058   1;
%e A276058   1;
%e A276058   2;
%e A276058   1,2;
%p A276058 G := (1-z^2)*(1-z+z^3)/(1-z-z^2+z^3-z^6+z^7-2*t*z^7): 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
%Y A276058 Cf. A078012, A226916, A276059.
%K A276058 nonn,tabf
%O A276058 0,5
%A A276058 _Emeric Deutsch_, Aug 22 2016