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

1,3

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

Table of n, a(n) for n=1..55.

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

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.

Sequence in context: A339033 A327028 A271707 * A360048 A127899 A128615

Adjacent sequences: A341442 A341443 A341444 * A341446 A341447 A341448

Keyword

nonn,tabl

Author

Sergi Elizalde, Feb 12 2021

Status

approved