OFFSET
1,3
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.
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).
EXAMPLE
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;
...
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 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Sergi Elizalde, Feb 12 2021
STATUS
approved