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A129154
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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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3
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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
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OFFSET
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0,7
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COMMENTS
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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.
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LINKS
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E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2192-2203.
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FORMULA
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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.
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EXAMPLE
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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;
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MAPLE
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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
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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