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A128747
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Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and having k peaks of height >1 (n >= 1; 0 <= k <= n-1).
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1
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1, 1, 2, 1, 7, 2, 1, 18, 15, 2, 1, 41, 68, 25, 2, 1, 88, 244, 171, 37, 2, 1, 183, 765, 866, 351, 51, 2, 1, 374, 2199, 3651, 2355, 636, 67, 2, 1, 757, 5954, 13601, 12708, 5421, 1058, 85, 2, 1, 1524, 15438, 46355, 58977, 36198, 11116, 1653, 105, 2, 1, 3059, 38747, 147768
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OFFSET
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1,3
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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.
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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) 2191-2203.
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FORMULA
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T(n,0) = 1.
Sum_{k=0..n-1} k*T(n,k) = A128748(n).
G.f.: G(t,z) = (1 - z + z*K(t,z))/(1 - z*K(t,z)) - 1, where K = K(t,z) satisfies zK^2 - (1 - tz)K + 1 - z = 0 (K is the g.f. for the number of peaks; see A126182).
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EXAMPLE
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T(3,1)=7 because we have UDU(UD)D, UDU(UD)L, U(UD)DUD, UU(UD)DD, UU(UD)LD, UU(UD)DL and UU(UD)LL (the peaks of height >1 are shown between parentheses).
Triangle starts:
1;
1, 2;
1, 7, 2;
1, 18, 15, 2;
1, 41, 68, 25, 2;
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MAPLE
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K:=(1-z*t-sqrt(z^2*t^2-2*z*t+1+4*z^2-4*z))/2/z: G:=z*(2*K-1)/(1-z*K): Gser:=simplify(series(G, z=0, 14)): for n from 1 to 11 do P[n]:=sort(coeff(Gser, z, n)) od: for n from 1 to 11 do seq(coeff(P[n], t, j), j=0..n-1) 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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