A339754 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having k symmetric vertices (n >= 1, 1 <= k <= n).

1, 0, 2, 0, 2, 3, 0, 2, 6, 6, 0, 4, 12, 16, 10, 0, 8, 24, 40, 40, 20, 0, 20, 60, 104, 120, 90, 35, 0, 50, 150, 270, 350, 330, 210, 70, 0, 140, 420, 768, 1040, 1080, 840, 448, 126, 0, 392, 1176, 2184, 3080, 3468, 3108, 2128, 1008, 252

Offset: 1

Author

Sergi Elizalde, Feb 12 2021

Keywords

• nonn
• tabl

Comments

A symmetric vertex is a vertex in the first half of the path (not including the midpoint) that is a mirror image of a vertex in the second half, with respect to reflection along the vertical line through the midpoint of the path.

Examples

			For n=5 there are 4 Dyck paths with 2 symmetric vertices: uuuuddddud, uduuuudddd, uuudddudud, ududuuuddd.
Triangle begins:
  1;
  0,   2;
  0,   2,    3;
  0,   2,    6,    6;
  0,   4,   12,   16,   10;
  0,   8,   24,   40,   40,   20;
  0,  20,   60,  104,  120,   90,   35;
  0,  50,  150,  270,  350,  330,  210,   70;
  0, 140,  420,  768, 1040, 1080,  840,  448,  126;
  0, 392, 1176, 2184, 3080, 3468, 3108, 2128, 1008, 252;
  ...
		

Links

• Alois P. Heinz, Rows n = 1..200, flattened
• Sergi Elizalde, The degree of symmetry of lattice paths, arXiv:2002.12874 [math.CO], 2020.
• Sergi Elizalde, Measuring symmetry in lattice paths and partitions, Sem. Lothar. Combin. 84B.26, 12 pp (2020).

Crossrefs

Row sums give A000108.

Main diagonal gives A001405.

Column k=2 gives 2*A005817(n-3) (for n>2).

Programs

Maple

b:= proc(x, y, v) option remember; expand(
      `if`(min(y, v, x-max(y, v))<0, 0, `if`(x=0, 1, (l-> add(add(
      `if`(y+i=v+j, z, 1)*b(x-1, y+i, v+j), i=l), j=l))([-1, 1]))))
    end:
g:= proc(n) option remember; add(b(n, j$2), j=0..n) end:
T:= (n, k)-> coeff(g(n), z, k):
seq(seq(T(n, k), k=1..n), n=1..10);  # Alois P. Heinz, Feb 12 2021

Mathematica

b[x_, y_, v_] := b[x, y, v] = Expand[If[Min[y, v, x - Max[y, v]] < 0, 0, If[x == 0, 1, Function[l, Sum[Sum[If[y + i == v + j, z, 1]*b[x - 1, y + i, v + j], {i, l}], {j, l}]][{-1, 1}]]]];
g[n_] := g[n] = Sum[b[n, j, j], {j, 0, n}];
T[n_, k_] := Coefficient[g[n], z, k];
Table[Table[T[n, k], {k, 1, n}], {n, 1, 10}] // Flatten (* Jean-François Alcover, Feb 13 2021, after Alois P. Heinz *)