Enumeration of lattice paths of a specific type

One of the approaches to "Special" meanders led (in particular) to the following question:

What is the number $$a_{m,n}(\ell)$$ of $$\ell$$-step paths from $$(1,1)$$ to $$(m,n)$$ using the following four kinds of steps: $$(i,j)\mapsto$$ $$(i+1,j)$$, $$(i+1,j-1)$$, $$(i,j+1)$$ or $$(i-1,j+1)$$ and with the restriction that $$i$$ and $$j$$ stay positive at each step?

The only thing I managed to find out is sort of a trivial reformulation of the definition: the polynomials $$P_\ell(x,y):=\sum_{m,n\geqslant1}a_{m,n}(\ell)x^my^n$$ satisfy a recurrence: since $$P_\ell(x,0)=P_\ell(0,y)=0$$, each $$F_\ell(x,y):=(x+y+x/y+y/x)P_\ell(x,y)$$ is a polynomial, and we have $$P_{\ell+1}(x,y)=F_\ell(x,y)-F_\ell(x,0)-F_\ell(0,y).$$

In a way of example - here is the table of the $$a_{m,n}(7)$$: $$\begin{array}{cccccccc} 0 & 1 & 21 & 80 & 125 & 85 & 21 & 1 \\ 1 & 28 & 139 & 254 & 210 & 76 & 7 & 0 \\ 21 & 139 & 306 & 308 & 140 & 21 & 0 & 0 \\ 80 & 254 & 308 & 168 & 35 & 0 & 0 & 0 \\ 125 & 210 & 140 & 35 & 0 & 0 & 0 & 0 \\ 85 & 76 & 21 & 0 & 0 & 0 & 0 & 0 \\ 21 & 7 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array}$$

As literature on the subject of lattice paths is way too vast for me to handle, this is mostly a reference request: I believe that working hardly enough I could find at least a generating function, but I also believe it must be done somewhere already.

In fact a more (or less?) specific question in this direction is whether there is some kind of searchable database of combinatorial objects where I could find such things. There is a question Combinatorial Databases here on MO but I could not find anything about path enumeration in the answers. The closest I could get is the Dyck path enumeration on FindStat

It seems the first solution to this problem appeared in Theorem 4 of

Raschel, Kilian, Counting walks in a quadrant: a unified approach via boundary value problems, J. Eur. Math. Soc. (JEMS) 14, No. 3, 749-777 (2012). ZBL1238.05014.

Notice that the walk corresponds to the fourth singular walk of Figure 2. The kernel $$K(x,y;z)$$ of the walk is given by $$K(x,y;z) = x y z(x +x/y +y +y/x -1/z).$$ Let $$X_0(y) = X_0(y;z)$$ (resp. $$Y_0(x)=Y_0(x;z)$$) be one of the solutions $$x$$ (resp. $$y$$) to $$K(x,y;z) = 0$$. Then Theorem 4 states that $$Q(x,y;z) = \sum_{m,n\geq 1} x^{m-1}y^{n-1} \sum_{\ell\geq 0} z^l a_{m,n}(\ell)$$ satisfies $$Q(x,y;z) = \frac{1}{K(x,y;z)}\left(z x^2Q(x,0;z)+z y^2 Q(0,y;z)-xy\right)$$ with $$Q(x,0;z) = \frac{1}{zx^2} \sum_{p\geq 0} Y_0 \circ (X_0\circ Y_0)^{\circ p}(x;z)\,\left[ (X_0\circ Y_0)^{\circ p}(x;z) - (X_0\circ Y_0)^{\circ (p+1)}(x;z) \right].$$ Here $$f^{\circ p}$$ means $$f\circ \cdots \circ f$$ with $$p$$ occurrences of $$f$$. Notice that by symmetry $$Q(0,y;z) = Q(y,0;z)$$.

A quick check with Mathematica reproduces the table for $$\ell=7$$:

k = x y z (x + x/y + y + y/x - 1/z);
x0[y_] = x /. Solve[k == 0, x][[1]] // FullSimplify[#, y > 0 && x > 0] &;
y0[x_] = y /. Solve[k == 0, y][[1]] // FullSimplify[#, y > 0 && x > 0] &;
Series[SeriesCoefficient[1/k (Sum[y0[xp] (xp - x0[y0[xp]]) /.
xp -> Nest[x0[y0[#]] &, x, p], {p, 0, 2}]
// -x y + # + (# /. x -> y) &), {z, 0, 7}], {x, 0, 7}, {y, 0, 7}]


yields the output

$$\left(y+21 y^2+80 y^3+125 y^4+85 y^5+21 y^6+y^7+O\left(y^8\right)\right)+x \left(1+28 y+139 y^2+254 y^3+210 y^4+76 y^5+7 y^6+O\left(y^8\right)\right)+x^2 \left(21+139 y+306 y^2+308 y^3+140 y^4+21 y^5+O\left(y^8\right)\right)+x^3 \left(80+254 y+308 y^2+168 y^3+35 y^4+O\left(y^8\right)\right)+x^4 \left(125+210 y+140 y^2+35 y^3+O\left(y^8\right)\right)+x^5 \left(85+76 y+21 y^2+O\left(y^8\right)\right)+x^6 \left(21+7 y+O\left(y^8\right)\right)+x^7+O\left(x^8\right)$$

• Indeed, I've corrected it. – Timothy Budd Jan 9 '19 at 20:32

This is an extension of the accepted answer. Using it, and ideas from some other papers, I managed to obtain an alternative expression for the generating function which I find interesting.

As explained in that answer, the problem reduces to finding $$Q(x,0,z):=\sum_{i,\ell}a_{n,1}(\ell)x^nz^\ell$$.

Let $$z=\frac1{q+q^{-1}}$$ and $$x=-\frac{(1-q) (1-q^2) t}{(1-t) (1-q t)q}$$. Then, $$x^2zQ(x,0,z)=-x+\frac{1-4z^2}{z^2}\sum _{k=1}^\infty \frac{(-1)^kq^kt}{\left(1-q^kt\right)^2}.$$

The series $$\sum _{k=1}^\infty \frac{(-1)^kq^kt}{\left(1-q^kt\right)^2}$$ seems to be related to elliptic functions, Jacobi forms and such stuff, but I cannot figure out how exactly. Maybe I will ask a separate question about it.

Few words about the proof. The expression for the generating function in the other answer can be equivalently described as follows. Suppose $$x$$, $$y$$ satisfy $$(x^2+y^2+x^2y+xy^2)z=xy$$. Then, $$(*)\qquad\qquad\qquad\qquad\qquad\qquad x^2zQ(x,0,z)=xy-y^2zQ(y,0,z).\qquad\qquad\qquad\qquad\qquad\qquad$$ Now the curve $$(x^2+y^2+x^2y+xy^2)z=xy$$ admits rational parametrizations $$x(t)$$, $$y(t)$$; moreover it is possible to choose such parametrization that satisfies $$y(t)=x(qt)$$, with $$z=\frac1{q+q^{-1}}$$ and $$x(t)=-\frac{(1-q) (1-q^2) t}{(1-t) (1-q t)q}$$. Then, denoting $$F(t,q)=\frac{x(t)^2}{q+q^{-1}}Q\left(x(t),0,\frac1{q+q^{-1}}\right)$$, we get from $$(*)$$ above the following functional equation for $$F$$: $$F(t,q)=\frac{(1-q)^2 \left(1-q^2\right)^2 t^2}{(1-t) (1-q t)^2 \left(1-q^2 t\right)q}-F(qt,q).$$ This leads more or less straightforwardly to the claimed equality.

Update

Expression in terms of "original" variables can be also derived from this. Although less pretty, it is, I believe, still more explicit than the one in terms of iterated radicals.

We have $$x^2zQ(x,0,z)=-x+\frac{1-4z^2}{z^2}\sum _{k=1}^\infty\frac{(-1)^k}{A(x,z)T_k\left(\frac{1}{2 z}\right)+B(x,z)U_{k-1}\left(\frac{1}{2 z}\right)-2},$$ where $$T$$ and $$U$$ are the Chebyshev polynomials, while $$A(x,z)$$ and $$B(x,z)$$ are quadratic irrationalities determined by \begin{aligned} A+\frac{2z}{1+2z}B&=2\left(1-\frac{1-2 z}{x z}\right),\\ A^2-\frac{4z^2}{1-4z^2}B^2&=4 \end{aligned}

Bousquet-Melou's work might be pertinent. See https://arxiv.org/abs/1708.06192 Figure 4 there seems to be your lattice walk.

• I meant for the answer above to be a comment. Sorry. – user61318 Jan 9 '19 at 13:48
• Only now I noticed, sorry - that case is different: it is (NE,SW,S,E) while this one is (NW,SE,N,E); since we are confined to the NE quadrant, the behaviour differs essentially - for example, in our situation we can never return to the origin while in that case one may revisit the origin arbitrarily often – მამუკა ჯიბლაძე Feb 9 '19 at 8:07