A132890 Triangle read by rows: T(n,k) is the number of left factors of Dyck paths of length n that have height k (1 <= k <= n).
1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 3, 4, 1, 1, 1, 7, 5, 5, 1, 1, 1, 7, 13, 6, 6, 1, 1, 1, 15, 18, 20, 7, 7, 1, 1, 1, 15, 39, 26, 27, 8, 8, 1, 1, 1, 31, 57, 73, 35, 35, 9, 9, 1, 1, 1, 31, 112, 99, 109, 44, 44, 10, 10, 1, 1, 1, 63, 169, 253, 152, 154, 54, 54, 11, 11, 1, 1
Offset: 1
Examples
T(5,3)=4 because we have UDUUU, UUDUU, UUUDD and UUUDU, where U=(1,1) and D=(1,-1). Triangle starts: 1; 1, 1; 1, 1, 1; 1, 3, 1, 1; 1, 3, 4, 1; 1; 1, 7, 5, 5, 1, 1;
Links
- Alois P. Heinz, Rows n = 1..141, flattened
- Steven R. Finch, How far might we walk at random?, arXiv:1802.04615 [math.HO], 2018.
- R. Kemp, On the average depth of a prefix of the Dycklanguage D_1, Discrete Math., 36, 1981, 155-170.
- Toufik Mansour, Gokhan Yilidirim, Longest increasing subsequences in involutions avoiding patterns of length three, Turkish Journal of Mathematics (2019), Section 2.2
Programs
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Maple
v := ((1-sqrt(1-4*z^2))*1/2)/z: g := proc (k) options operator, arrow: v^k*(1+v)*(1+v^2)/((1+v^(k+1))*(1+v^(k+2))) end proc: T := proc (n, k) options operator, arrow: coeff(series(g(k), z = 0, 50), z, n) end proc: for n from 0 to 12 do seq(T(n, k), k = 1 .. n) end do; # yields sequence in triangular form # second Maple program: b:= proc(x, y, k) option remember; `if`(x=0, z^k, `if`(y>0, b(x-2, y-1, k), 0)+ b(x-2, y+1, max(y+1, k))) end: T:= n-> (p-> seq(coeff(p, z, i), i=1..n))(b(2*n, 0$2)): seq(T(n), n=1..16); # Alois P. Heinz, Sep 05 2017 -
Mathematica
b[x_, y_, k_] := b[x, y, k] = If[x == 0, z^k, If[y > 0, b[x - 2, y - 1, k], 0] + b[x - 2, y + 1, Max[y + 1, k]]]; T[n_] := Function[p, Table[Coefficient[p, z, i], {i, 1, n}]][b[2n, 0, 0]]; Table[T[n], {n, 1, 16}] // Flatten (* Jean-François Alcover, Apr 01 2018, after Alois P. Heinz *)
Formula
The g.f. of column k is g(k, z) = v^k*(1+v)*(1+v^2)*/((1+v^(k+1))*(1+v^(k+2))), where v = (1-sqrt(1-4*z^2))/(2*z). (Obtained as the difference G(k,z)-G(k-1,z), where G(k,z) is given in the R. Kemp reference (p. 159).)
Extensions
Keyword tabl added by Michel Marcus, Apr 09 2013
Comments