A341445 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having degree of symmetry k (n >= 1, 1 <= k <= n).

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

Offset: 1

Author

Sergi Elizalde, Feb 12 2021

Keywords

• nonn
• tabl

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.

Examples

			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;
  ...
		

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).

Crossrefs

Equivalent to A298645 with rows reversed.

Row sums give A000108.

Main diagonal gives A001405.

Column k=1 gives A298647 (for n>2).

Second subdiagonal gives 2*A191522.

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=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 *)