A276058 Triangle read by rows: T(n,k) is the number of compositions of n with parts in {3,4,5,6,...} and having asymmetry degree equal to k (n>=0; 0<=k<=floor(n/7)).
1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 4, 3, 6, 3, 10, 5, 14, 4, 24, 7, 30, 4, 6, 46, 8, 10, 58, 20, 9, 84, 36, 15, 106, 68, 13, 152, 112, 22, 188, 196, 19, 264, 304, 8, 32, 324, 492, 24, 28, 446, 732, 72, 47, 546, 1120, 160, 41, 744, 1616, 344
Offset: 0
Examples
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;
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.
Programs
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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
Formula
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]=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.
Comments