A129172 Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n such that the area between the x-axis and the path is k (n >= 0, 0 <= k <= n^2).

1, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 3, 2, 1, 1, 1, 0, 0, 0, 0, 1, 1, 4, 5, 5, 3, 5, 4, 3, 2, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 5, 7, 12, 10, 11, 12, 14, 12, 10, 8, 10, 7, 5, 4, 3, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 6, 9, 19, 23, 28, 26, 36, 38, 38, 32, 36, 36, 34, 29, 27, 25, 21, 15, 16, 13, 10, 7, 5

Offset: 0

Author

Emeric Deutsch, Apr 09 2007

Keywords

• nonn
• tabf

Comments

A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1)(up), D=(1,-1)(down) and L=(-1,-1)(left) so that up and left steps do not overlap. The length of the path is defined to be the number of its steps.

Row n has n^2 + 1 terms, the first n of which are equal to 0.

Row sums yield A002212.

Sum of terms in column k is the Fibonacci number F(k) (k >= 1; F(1)=1, F(2)=1; A000045).

Examples

			T(4,7)=5 because we have UDUUUDLD, UDUUDUDL, UUDDUUDL, UUUDLDUD and UUUUDLLL.
Triangle starts:
  1;
  0, 1;
  0, 0, 1, 1, 1;
  0, 0, 0, 1, 1, 3, 2, 1, 1, 1;
  0, 0, 0, 0, 1, 1, 4, 5, 5, 3, 5, 4, 3, 2, 1, 1, 1;
		

Links

• Alois P. Heinz, Rows n = 0..40, flattened
• E. Deutsch, E. Munarini and S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203.
• Emeric Deutsch, Emanuele Munarini and Simone Rinaldi, Skew Dyck paths, area, and superdiagonal bargraphs, Journal of Statistical Planning and Inference, Vol. 140, Issue 6, June 2010, pp. 1550-1562.

Crossrefs

Cf. A000045, A002212, A129173, A274372.

Programs

Maple

G:=(1-z+z*g[1])/(1-t*x*z*g[1]): for i from 1 to 9 do g[i]:=(1-z+z*g[i+1])/(1-t^(2*i+1)*x*z*g[i+1]) od: g[10]:=0: x:=1: Gser:=simplify(series(G,z=0,9)): for n from 0 to 7 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 0 to 7 do seq(coeff(P[n],t,j),j=0..n^2) od; # yields sequence in triangular form
# second Maple program:
b:= proc(n, y, t) option remember; expand(`if`(y>n, 0,
      `if`(n=0, 1, `if`(t<0, 0, b(n-1, y+1, 1)*z^(y+1/2))+
      `if`(y<1, 0, b(n-1, y-1, 0)*z^(y-1/2))+
      `if`(t>0 or y<1, 0, b(n-1, y-1, -1)*z^(1/2-y)))))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..n^2))(b(2*n, 0$2)):
seq(T(n), n=0..8);  # Alois P. Heinz, Jun 19 2016

Mathematica

b[n_, y_, t_] := b[n, y, t] = Expand[If[y > n, 0, If[n == 0, 1, If[t < 0, 0, b[n - 1, y + 1, 1]*z^(y + 1/2)] + If[y < 1, 0, b[n - 1, y - 1, 0]*z^(y - 1/2)] + If[t > 0 || y < 1, 0, b[n - 1, y - 1, -1]*z^(1/2 - y)]]]]; T[n_] := Function[p, Table[Coefficient[p, z, i], {i, 0, n^2}]][b[2*n, 0, 0]]; Table[T[n], {n, 0, 8}] // Flatten (* Jean-François Alcover, Dec 20 2016, after Alois P. Heinz *)

Formula

Sum_{k=0..n^2} k*T(n,k) = A129173(n).

G.f.: G(t,z) = H(t,1,z), where H(t,x,z) = 1+txzH(t,t^2*x,z)H(t,x,z) + z[H(t,t^2*x,z)-1] (H(t,x,z) is the trivariate g.f. for skew Dyck paths according to area, semiabscissa of the last point on the x-axis and semilength, marked by t,x and z, respectively).