A339754 - OEIS

%I #40 Feb 13 2021 19:38:13

%S 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,

%T 35,0,50,150,270,350,330,210,70,0,140,420,768,1040,1080,840,448,126,0,

%U 392,1176,2184,3080,3468,3108,2128,1008,252

%N 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).

%C 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.

%H Alois P. Heinz, <a href="/A339754/b339754.txt">Rows n = 1..200, flattened</a>

%H Sergi Elizalde, <a href="https://arxiv.org/abs/2002.12874">The degree of symmetry of lattice paths</a>, arXiv:2002.12874 [math.CO], 2020.

%H Sergi Elizalde, <a href="https://www.mat.univie.ac.at/~slc/wpapers/FPSAC2020/26.html">Measuring symmetry in lattice paths and partitions</a>, Sem. Lothar. Combin. 84B.26, 12 pp (2020).

%e For n=5 there are 4 Dyck paths with 2 symmetric vertices: uuuuddddud, uduuuudddd, uuudddudud, ududuuuddd.

%e Triangle begins:

%e   1;

%e   0,   2;

%e   0,   2,    3;

%e   0,   2,    6,    6;

%e   0,   4,   12,   16,   10;

%e   0,   8,   24,   40,   40,   20;

%e   0,  20,   60,  104,  120,   90,   35;

%e   0,  50,  150,  270,  350,  330,  210,   70;

%e   0, 140,  420,  768, 1040, 1080,  840,  448,  126;

%e   0, 392, 1176, 2184, 3080, 3468, 3108, 2128, 1008, 252;

%e   ...

%p b:= proc(x, y, v) option remember; expand(

%p       `if`(min(y, v, x-max(y, v))<0, 0, `if`(x=0, 1, (l-> add(add(

%p       `if`(y+i=v+j, z, 1)*b(x-1, y+i, v+j), i=l), j=l))([-1, 1]))))

%p     end:

%p g:= proc(n) option remember; add(b(n, j$2), j=0..n) end:

%p T:= (n, k)-> coeff(g(n), z, k):

%p seq(seq(T(n, k), k=1..n), n=1..10);  # _Alois P. Heinz_, Feb 12 2021

%t 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}]]]];

%t g[n_] := g[n] = Sum[b[n, j, j], {j, 0, n}];

%t T[n_, k_] := Coefficient[g[n], z, k];

%t Table[Table[T[n, k], {k, 1, n}], {n, 1, 10}] // Flatten (* _Jean-François Alcover_, Feb 13 2021, after _Alois P. Heinz_ *)

%Y Row sums give A000108.

%Y Main diagonal gives A001405.

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

%K nonn,tabl

%O 1,3

%A _Sergi Elizalde_, Feb 12 2021