A316728 Number T(n,k) of permutations of {0,1,...,2n} with first element k whose sequence of ascents and descents forms a Dyck path; triangle T(n,k), n>=0, 0<=k<=2n, read by rows.

1, 1, 1, 0, 8, 7, 5, 2, 0, 172, 150, 121, 87, 52, 22, 0, 7296, 6440, 5464, 4411, 3337, 2306, 1380, 604, 0, 518324, 463578, 405024, 344260, 283073, 223333, 166856, 115250, 69772, 31238, 0, 55717312, 50416894, 44928220, 39348036, 33777456, 28318137, 23068057, 18117190, 13543456, 9409366, 5759740, 2620708, 0

Offset: 0

Author

Alois P. Heinz, Jul 11 2018

Keywords

• nonn
• tabf

Examples

			T(2,0) = 8: 01432, 02143, 02431, 03142, 03241, 03421, 04132, 04231.
T(2,1) = 7: 12043, 12430, 13042, 13240, 13420, 14032, 14230.
T(2,2) = 5: 23041, 23140, 23410, 24031, 24130.
T(2,3) = 2: 34021, 34120.
T(2,4) = 0.
Triangle T(n,k) begins:
       1;
       1,      1,      0;
       8,      7,      5,      2,      0;
     172,    150,    121,     87,     52,     22,      0;
    7296,   6440,   5464,   4411,   3337,   2306,   1380,    604,     0;
  518324, 463578, 405024, 344260, 283073, 223333, 166856, 115250, 69772, 31238, 0;
		

Links

• Alois P. Heinz, Rows n = 0..100, flattened
• Wikipedia, Counting lattice paths

Crossrefs

Column k=0 gives A303285.

Row sums and T(n+1,2n+1) give A177042.

T(n,n) gives A316727.

T(n+1,n) gives A316730.

T(n,2n) gives A000007.

Cf. A079484.

Programs

Maple

b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
      `if`(t>0,   add(b(u-j, o+j-1, t-1), j=1..u), 0)+
      `if`(o+u>t, add(b(u+j-1, o-j, t+1), j=1..o), 0))
    end:
T:= (n, k)-> b(k, 2*n-k, 0):
seq(seq(T(n, k), k=0..2*n), n=0..8);

Mathematica

b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, 1,
     If[t > 0,     Sum[b[u - j, o + j - 1, t - 1], {j, 1, u}], 0] +
     If[o + u > t, Sum[b[u + j - 1, o - j, t + 1], {j, 1, o}], 0]];
T[n_, k_] := b[k, 2n - k, 0];
Table[Table[T[n, k], {k, 0, 2n}], {n, 0, 8}] // Flatten (* Jean-François Alcover, Mar 27 2021, after Alois P. Heinz *)

Formula

Sum_{k=0..2n} T(n,k) = T(n+1,2n+1) = A177042(n).

Sum_{k=0..2n} (k+1) * T(n,k) = A079484(n).