A129155 -id:A129155 - cp's OEIS frontend

A129154 Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and having k primitive Dyck factors (n >= 0; 0 <= k <= n).

1, 0, 1, 1, 1, 1, 4, 3, 2, 1, 15, 11, 6, 3, 1, 59, 41, 22, 10, 4, 1, 241, 159, 84, 38, 15, 5, 1, 1011, 639, 331, 150, 60, 21, 6, 1, 4326, 2640, 1342, 606, 246, 89, 28, 7, 1, 18797, 11146, 5570, 2500, 1023, 380, 126, 36, 8, 1, 82685, 47884, 23567, 10503, 4312, 1630

Offset: 0

Emeric Deutsch, Apr 02 2007

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. A primitive Dyck factor is a subpath of the form UPD that starts on the x-axis, P being a Dyck path.

Row sums yield A002212.

			T(4,2)=6 because we have (UD)(UD)UUDL, (UD)(UUDUDD), (UD)(UUUDDD), (UUDD)(UUDD), (UUDUDD)(UD) and (UUUDDD)(UD) (the primitive Dyck factors are shown between parentheses).
Triangle starts:
   1;
   0,  1;
   1,  1,  1;
   4,  3,  2,  1;
  15, 11,  6,  3,  1;

E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2192-2203.

Cf. A002212, A129155, A129156.

G:=(3-3*z-sqrt(1-6*z+5*z^2))/(2-t+z+(t-1)*sqrt(1-4*z)+sqrt(1-6*z+5*z^2)): Gser:=simplify(series(G,z=0,14)): for n from 0 to 11 do P[n]:=sort(coeff(Gser,z,n)) od: for n from 0 to 11 do seq(coeff(P[n],t,j),j=0..n) od; # yields sequence in triangular form

T(n,0) = A129155(n).
Sum_{k=0..n} k*T(n,k) = A129156(n).
G.f.: G(t,z) = (1 + z(g-1))/(1 - z(g-C) - tzC), where g = 1 + zg^2 + z(g-1) = (1 - z - sqrt(1 - 6z + 5z^2))/(2z) and C = 1 + zC^2 = (1-sqrt(1-4z))/(2z) is the Catalan function.

cp's OEIS Frontend

A129154 Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and having k primitive Dyck factors (n >= 0; 0 <= k <= n).

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