OFFSET
0,5
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).
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*T(n,k), k>=0)=A276059(n).
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.
FORMULA
G.f.: G(t,z) = (1-z^2)(1-z +z^3)/(1-z-z^2+z^3-z^6+z^7-2tz^7). In the more general situation of compositions into a[1]<a[2]<a[3]<..., denoting F(z) = Sum(z^{a[j]},j>=1}, 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.
EXAMPLE
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,
MAPLE
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
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Aug 22 2016
STATUS
approved