‎Let ‎$‎n‎$ ‎be a‎ ‎positive ‎integers and ‎$‎T=T_{n,n}‎$ ‎be the ‎$‎n\times n‎$‎ table in the first quadrant composed of $n^2$ unit squares‎, ‎whose $(x,y)$-blank is locate in the $x^{th}$-column from the left and the $y^{th}$-row from the bottom hand side of $T_{n,n}$ . ‎ Put ‎$‎D(n,n)‎$ ‎be ‎the ‎number ‎of ‎all ‎lattice ‎path ‎from the ‎first ‎column to entry ‎$‎(n,n)‎$‎ ‎of the ‎table ‎‎$‎T‎$ which steps comes from the set $S=\{(1,0)‎, ‎(1,1),(1,-1)\}$.(we allowed to move only to the right (up, down or straight) ). ‎It ‎is ‎easy ‎to ‎see ‎for ‎‎$‎n\geq 2‎$‎ $$D(n,n)=D(n-1,n)+D(n-1,n-1)$$ ‎where ‎$‎D(1,1)=1, D(2,2)=2, D(3,3)=5,‎ D(4,4)=13, \cdots‎‎$‎.

Notice, the entry $(x,y)$ means cordinate $x$ and $y$ in the table $T$ not the row $x$ and column $y$.

For example $$D(3,3)=D(2,3)+D(2,2)=2+3=5$$ and $$D(2,3)=D(1,3)+D(1,2)=1+1=2.$$ For calculating $D(2,3)$ you must consider the table with 3 rows and columns and by using this table calculate all lattice paths reach to entry $(2,3)$ in this table!! in my arXiv paper, there are some references for this sequence!! I ‎think ‎these ‎lattice ‎paths ‎very ‎interesting ‎and ‎obtained ‎some ‎results ‎about ‎them. Put ‎$‎D(n,n)=d_n‎‎$, I check and known that the Hankel determinant evaluation of ‎$‎D(n,n)‎$ is

$$ \det(H_n^1)=\det‎‎ \begin{bmatrix} d_{1} & d_{2} & d_{3} & \dots & d_{n} \\ d_{2} & d_{3} & d_{4} & \dots & d_{n+1} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ d_{n} & d_{n+1} & d_{n+2} & \dots & d_{2n-1} \end{bmatrix}‎‎ =1. $$‎‎‎‎ Do you have ideas or comments for proving it?

Thank you so much for any help or comment.

  • $\begingroup$ What is $D(n-1, n+1)$? You seem only to have defined $D(n,n)$. If it means what it seems to mean then how can your relation for $D(n,n)$ involve $D(n-1,n+1)$ as that would seem to involve your path leaving the square $T_{n,n}$. $\endgroup$ Aug 14 '17 at 21:45
  • $\begingroup$ $D(n,n)$ is the number of all lattice paths from the first column to the entry $(n,n)$, then, $$D(n,n)=D(n-1,n)+D(n-1,n-1)$$. $\endgroup$
    – d.y
    Aug 14 '17 at 21:51
  • $\begingroup$ You have not read my comment. You have not said what $D(n-1, n+1)$ is. The relation you have written in your comment is not the same as the relation you have written in the question. $\endgroup$ Aug 14 '17 at 21:56
  • $\begingroup$ I check the relation in question and fixed it. actually, if we have a table with $n$ rows and $m$ columns and let $D(x,y)$ denoted the number of lattice paths from first column to the entra$(x,y)$ we have $$D(x+1,y)=D(x,y-1)+D(x,y)+D(x,y+1)$$ where $D(x,0)=D(x,m+1)=0$ for all $x\geq 1$ and $y=1,2,\cdots , m$. In the above qustion I consider for specail case $m=n$. Also, $D(n-1,n+1)$ is the number of lattice paths with three steps $(1,1),(1,-1),(1,0)$ from the first columns to the entry $(n-1,n+1)$. $\endgroup$
    – d.y
    Aug 14 '17 at 22:05
  • $\begingroup$ You've now changed you relation in the question so it doesn't involve $D(n-1, n+1)$. Can you explain what $D(2,3)$ is? You have not defined it. With the definition I would guess it seems to be 2. That would mean that $D(3,3) = 2 + 2 =4 \ne 5$. $\endgroup$ Aug 14 '17 at 22:07

The $d(n):=D(n,n)$ is OEIS sequence A005773. The Hankel determinant property is given in the sequence entry. Also the recursion $\;nd_{n}=2nd_{n-1}+3(n-2)d_{n-2}$. A proof could come from a similar proof of the Hankel determinant property of the Catalan numbers. A related property is to use $d_0=1$ and take $\det(H_n^0)$ with $d_0$ in the first row and column giving OEIS sequence A163806 $(1,1,1,0,-1,-1,0,1,1,0,-1,-1,0,1,1,\dots)$ with a period of $6$ after the initial $d_0$.

The proof of the Hankel properties could probably use the Lindstrom-Gessel-Viennot Lemma. This can be used to show that the unique solution to $\det(H_n^0)=\det(H_n^1)=1$ for all $n$ is the Catalan numbers. For a reference look at Aigner, Catalan-like numbers and determinants, J. Comb. Th. A 87 (1999), 33-51.

By the way, notice how the diagram for $D(4,4)=13$ is $$ \begin{matrix} 1 & 2 & 5 & 13 \\ 1 & 3 & 8 & 21 \\ 1 & 3 & 8 & 21 \\ 1 & 2 & 5 & 13 \end{matrix}‎‎$$ where each entry is the sum of two or three entries in the preceding column.

Catalan and related sequence proofs are given by J, Cigler in Some nice Hankel determinants.

  • $\begingroup$ How proof the Hankel determinant of the Catalan numbers? I can proof the recursion $\;nd_{n}=2nd_{n-1}+3(n-2)d_{n-2}$ with generating function of $D(n,n)$,but I have no combinatorial proof for it? Do you have idea for combinatorial proof this recursion relation? $\endgroup$
    – d.y
    Aug 14 '17 at 23:43
  • $\begingroup$ Such things are discussed in Christian Krattenthaler's "Advanced determinant calculus" (arxiv.org/abs/math/9902004 & arxiv.org/abs/math/0503507). $\endgroup$ Aug 15 '17 at 9:40
  • $\begingroup$ @Wadim Zudilin I see "Advanced determinant calculus", but I couldn't find the proof of Hankel determinant of Catalan numbers. $\endgroup$
    – d.y
    Aug 15 '17 at 16:58

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