0,5

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).

A sequence is palindromic if and only if its asymmetry degree i 0.

Number of entries in row n is 1 + floor(n/7).

Sum of entries in row n is A078012(n).

T(n,0) = A226916(n+4)

Sum_{k>=0} k*T(n,k) = A276059(n).

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

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.

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.

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.

Triangle starts:

1;

0;

0;

1;

1;

1;

2;

1,2;

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

Cf. A078012, A226916, A276059.

Sequence in context: A241314 A276056 A276060 * A247302 A029252 A094876

Adjacent sequences: A276055 A276056 A276057 * A276059 A276060 A276061

nonn,tabf

Emeric Deutsch, Aug 22 2016

approved