A276062 Triangle read by rows: T(n,k) is the number of compositions of n with parts in {1,4} and having asymmetry degree equal to k (n>=0; 0<=k<=floor(n/4)).

1, 1, 1, 1, 2, 1, 2, 2, 2, 1, 4, 3, 4, 2, 8, 4, 6, 4, 3, 12, 4, 6, 8, 12, 4, 20, 12, 8, 14, 28, 5, 32, 24, 8, 11, 24, 52, 8, 7, 52, 40, 32, 15, 38, 96, 32, 10, 80, 72, 88, 21, 56, 172, 80, 16, 14, 122, 132, 192, 16, 29, 84, 304, 160, 80, 19, 184, 232, 392, 80, 40, 128, 516, 312, 256, 26, 278, 384, 768, 240, 32

Offset: 0

Author

Emeric Deutsch, Aug 22 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/5).

Sum of entries in row n is A003269(n+1).

T(n,0) = A226517(n+10)

Sum_{k>=0} k*T(n,k) = A276063(n).

Examples

			Row 6 is [2,2] because the compositions of 6 with parts in {1,4} are 411, 141, 114, and 111111, having asymmetry degrees 1, 0, 1, and 0, respectively.
Triangle starts:
  1;
  1;
  1;
  1;
  2;
  1, 2;
  2, 2;
  1, 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. A003269, A226517, A276063.

Programs

Maple

G := (1+z+z^4)/(1-z^2-2*t*z^5-z^8): 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 == 4]]], 1]]], {n, 0, 25}] // Flatten (* Michael De Vlieger, Aug 22 2016 *)

Formula

G.f.: G(t,z) = (1+z+z^4)/(1-z^2-2*t*z^5-z^8). 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.