cp's OEIS Frontend

A276064 Triangle read by rows: T(n,k) is the number of compositions of n with parts in {1,5} and having asymmetry degree equal to k (n>=0; 0<=k<=floor(n/6)).

%I A276064 #18 Apr 12 2025 05:17:46
%S A276064 1,1,1,1,1,2,1,2,2,2,1,4,2,4,2,6,3,8,3,8,4,4,12,4,4,10,12,6,16,12,5,
%T A276064 16,24,8,24,28,6,26,40,8,10,36,52,8,8,40,60,32,13,56,84,32,11,58,96,
%U A276064 80,17,84,136,88,15,80,160,160,16,23,120,220,192,16
%N A276064 Triangle read by rows: T(n,k) is the number of compositions of n with parts in {1,5} and having asymmetry degree equal to k (n>=0; 0<=k<=floor(n/6)).
%C A276064 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 A276064 Number of entries in row n is 1 + floor(n/6).
%C A276064 Sum of entries in row n is A003520(n).
%C A276064 T(n,0) = A226516(n+11).
%C A276064 Sum_{k>=0} k*T(n,k) = A276065(n).
%D A276064 S. Heubach and T. Mansour, Combinatorics of Compositions and Words, CRC Press, 2010.
%H A276064 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 A276064 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 A276064 G.f.: G(t,z) = (1+z+z^5)/(1-z^2-2*t*z^6-z^10). 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 A276064 Row 8 is [1,4] because the compositions of 8 with parts in {1,5} are 5111, 1511, 1151, 1115 and 11111111, having asymmetry degrees 1, 1, 1, 1, and 0, respectively.
%e A276064 Triangle starts:
%e A276064   1;
%e A276064   1;
%e A276064   1;
%e A276064   1;
%e A276064   1;
%e A276064   2;
%e A276064   1, 2;
%e A276064   2, 2.
%p A276064 G := (1+z+z^5)/(1-z^2-2*t*z^6-z^10): 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 A276064 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 == 5]]], 1]]], {n, 0, 25}] // Flatten (* _Michael De Vlieger_, Aug 22 2016 *)
%Y A276064 Cf. A003520, A226516, A276065.
%K A276064 nonn,tabf
%O A276064 0,6
%A A276064 _Emeric Deutsch_, Aug 22 2016