A247285 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n (n>=1) having k (0<=k<=n-1) upper interactions.

1, 1, 1, 1, 3, 1, 1, 5, 7, 1, 1, 7, 19, 14, 1, 1, 9, 36, 59, 26, 1, 1, 11, 58, 150, 162, 46, 1, 1, 13, 85, 300, 543, 408, 79, 1, 1, 15, 117, 523, 1335, 1771, 966, 133, 1, 1, 17, 154, 833, 2747, 5303, 5335, 2184, 221, 1, 1, 19, 196, 1244, 5031, 12792, 19272, 15099, 4767, 364, 1

Offset: 1

Author

Emeric Deutsch, Sep 11 2014

Keywords

• nonn
• tabl

Comments

An upper interaction in a Dyck path is an occurrence of a string d^k u^k for some k>=1; here u = (1,1) and d = (1,-1). For example, the Dyck path uu[d(du)u]dd has 2 upper interactions, shown between parentheses.

Number of entries in row n is n.

Sum of entries in row n is the Catalan number A000108(n).

Sum(k*T(n,k), k>=0) = A172061(n-2).

The statistic "number of lower interactions", mentioned in the Le Borgne reference is basically identical with the statistic "pyramid weight" of the Denise and Simion reference (see A091866 and the bottom of p. 8 of the Le Borgne reference).

T(n+1,n) = A001924(n) for n>=1. - Alois P. Heinz, Sep 11 2014

Examples

			Row 3 is 1,3,1. Indeed, the number of upper interactions in uuuddd, uududd, uuddud, uduudd, and ududud are 0, 1, 1, 1, and 2, respectively.
Triangle starts:
1;
1,1;
1,3,1;
1,5,7,1;
1,7,19,14,1;
1,9,36,59,26,1;
		

Links

• Alois P. Heinz, Rows n = 1..141, flattened
• A. Denise and R. Simion, Two combinatorial statistics on Dyck paths, Discrete Math., 137, 1995, 155-176.
• Y. Le Borgne, Counting upper interactions in Dyck paths, Sem. Lotharingien de Combinatoire, 54, 2006, Article B54f.

Crossrefs

Cf. A000108, A001924, A172061, A091866.

Programs

Maple

q := u*t: s := ((1+t-2*q-sqrt((1-t)*(1-t-4*q+4*q^2)))*(1/2))/(t*(1-q)): Q := proc (x, n) options operator, arrow: product(1-q^k*x, k = 0 .. n-1) end proc: A := -t*add(((q-t)*s/(1-q))^n*q^(binomial(n+2, 2)-1)/(Q(q, n)*Q(q*t*s^2, n)), n = 0 .. 15)/add(((q-t)*s/(1-q))^n*q^binomial(n+2, 2)*(1-t*q^n*s)/(Q(q, n)*Q(q*t*s^2, n)*(1-q^n*s)*(1-q^(n+1)*s)), n = 0 .. 15): Aser := simplify(series(A, t = 0, 22)): for n to 16 do P[n] := sort(coeff(Aser, t, n)) end do: for n to 13 do seq(coeff(P[n], u, j), j = 0 .. n-1) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(x, y, t, c) option remember; `if`(y<0 or y>x, 0,
     `if`(x=0, 1, expand(b(x-1, y+1, false, max(0, c-1))*
     `if`(c>0, z, 1)+b(x-1, y-1, true, 1+`if`(t, c, 0)))))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..n-1))(b(2*n, 0, false, 0)):
seq(T(n), n=1..15);  # Alois P. Heinz, Sep 11 2014

Mathematica

b[x_, y_, t_, c_] := b [x, y, t, c] = If[y<0 || y>x, 0, If[x == 0, 1, Expand[b[x-1, y+1, False, Max[0, c-1]]*If[c>0, z, 1] + b[x-1, y-1, True, 1 + If[t, c, 0] ] ] ] ]; T[n_] := Function[{p}, Table[Coefficient[p, z, i], {i, 0, n-1}]][b[2*n, 0, False, 0]]; Table[T[n], {n, 1, 25}] // Flatten (* Jean-François Alcover, May 27 2015, after Alois P. Heinz *)

Formula

The g.f. A(t,u), where t marks semilength and u marks upper interactions, is given in Proposition 2 of the Le Borgne reference. It is extremely complex; the Maple program follows it (blindly), except that the infinite sums have been replaced by summations from n=0 to n=15.