OFFSET
1,3
COMMENTS
The degree of symmetry of a Dyck path is defined as the number of steps in the first half that are mirror images of steps in the second half, with respect to the reflection along the vertical line through the midpoint of the path.
LINKS
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=4 there are 6 Dyck paths with degree of symmetry equal to 2: uuuddudd, uuduuddd, uududdud, uuddudud, uduududd, ududuudd.
Triangle begins:
1;
0, 2;
2, 0, 3;
2, 6, 0, 6;
8, 8, 16, 0, 10;
16, 32, 24, 40, 0, 20;
52, 84, 108, 60, 90, 0, 35;
134, 262, 294, 310, 150, 210, 0, 70;
432, 816, 1008, 880, 816, 336, 448, 0, 126;
1248, 2544, 3192, 3208, 2460, 2100, 784, 1008, 0, 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=v+(j-i)/2, 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 == v + (j - i)/2, 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