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)).

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, 36, 8, 9, 36, 52, 32, 15, 58, 76, 40, 13, 60, 108, 80, 16, 22, 96, 160, 112, 16, 19, 100, 204, 192, 80, 32, 160, 312, 272, 96, 28, 162, 376, 440, 240, 32

Offset: 0

Author

Emeric Deutsch, Aug 18 2016

Keywords

• nonn
• tabf

Comments

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).

Number of entries in row n is 1 + floor(n/4).

Sum of entries in row n is A000930(n).

Examples

			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.
Triangle starts:
  1;
  1;
  1;
  2;
  1, 2;
  2, 2;
  2, 4;
  ...
		

References

• S. Heubach and T. Mansour, Combinatorics of Compositions and Words, CRC Press, 2010.

Links

• Krithnaswami Alladi and V. E. Hoggatt, Jr. Compositions with Ones and Twos, Fibonacci Quarterly, 13 (1975), 233-239.
• V. E. Hoggatt, Jr. and Marjorie Bicknell, Palindromic compositions, Fibonacci Quart., Vol. 13(4), 1975, pp. 350-356.

Crossrefs

Cf. A000930, A226916, A276057.

Programs

Maple

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

Mathematica

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 *)

Formula

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]=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.

T(n,0) = A226916(n+7).

Sum_{k>=0} k*T(n,k) = A276057(n).