[Paper Review] Counting of paths and the multiplicity of determinantal rings

Hsin-Ju Wang|ArXiv.org|Oct 24, 2002

Advanced Combinatorial Mathematics6 references3 citations

TL;DR

This paper derives combinatorial formulas for counting non-intersecting lattice paths in two-sided ladder-shaped regions, using the Lindström–Gessel–Viennot lemma to compute the multiplicity of determinantal rings associated with ladder matrices. The key contribution is a new determinant formula for the multiplicity of ladder determinantal rings, extending classical results on determinantal ideals via path enumeration and sign-reversing involutions.

ABSTRACT

In this paper, we derive several formulas of counting families of non-intersecting paths for two-sided ladder-shaped regions. As an application, we give a new proof to a combinatorial interpretation of Fibonacci numbers obtained by G. Andrews in 1974.

Motivation & Objective

• To compute the multiplicity of ladder determinantal rings arising from minors of submatrices in a two-sided ladder-shaped region.
• To generalize classical multiplicity formulas for determinantal rings to the case of non-rectangular ladder-shaped matrices.
• To provide a combinatorial interpretation of the multiplicity via families of non-intersecting lattice paths.
• To establish a new formula for the multiplicity of Pfaffian rings associated with skew-symmetric matrices using path enumeration techniques.

Proposed method

• Use the Lindström–Gessel–Viennot lemma to count families of non-intersecting lattice paths in two-sided ladder-shaped regions.
• Define a partial order on lattice points to model paths from source to sink in a grid with constraints.
• Apply sign-reversing involutions to simplify sums over lattice paths with alternating signs.
• Use the determinant of a matrix of path counts to compute the number of non-intersecting path systems.
• Transform the original ring via Gröbner basis theory to a monomial ideal, linking multiplicity to the number of facets in a simplicial complex.
• Reduce the problem to counting non-intersecting paths in a subregion by removing redundant path components via a geometric transformation.

Experimental results

Research questions

• RQ1 How can the multiplicity of a ladder determinantal ring be computed using combinatorial path counting?
• RQ2 What is the structure of non-intersecting lattice paths in a two-sided ladder-shaped region of the integer lattice?
• RQ3 Can the multiplicity of Pfaffian rings associated with skew-symmetric matrices be expressed via a determinant of path counts?
• RQ4 How do sign-reversing involutions simplify the sum over lattice paths with alternating signs?
• RQ5 What is the precise form of the multiplicity formula for ladder determinantal rings with non-rectangular support?

Key findings

• The multiplicity of a ladder determinantal ring defined by $(r+1)$-minors of a ladder-shaped submatrix is given by a determinant of a matrix whose entries are signed sums of binomial coefficients over lattice paths.
• For a ladder $Y$ defined by $\{x_{ij} \mid 1 \leq i \leq m, \max\{1,l-i+2\} \leq j \leq \min\{n,n+m-k-i\}\}$, the multiplicity is $\det\left[\sum_{(a,b)\in T}(-1)^{a+b}\binom{m+n-i-j}{m-1+(a-b-1)(i-1)+a(k-m)+b(l-n)}\right]_{i,j=1}^r$, where $T = \{(a,b) \mid a-b=0 \text{ or } 1\}$.
• For a diagonal ladder in a skew-symmetric matrix, the multiplicity of the Pfaffian ring $K[Y]/Q_r$ is $\det\left[\sum_{(a,b)\in T}(-1)^{a+b}\binom{2n-2r-i-j}{n-r-i+a(-r+i-1)+b(r-l-i)}\right]_{i,j=1}^{r-1}$.
• In the special case $l = n-1$, the Pfaffian multiplicity simplifies to $\det\left[\binom{2n-2r-i-j}{n-r-i} - \binom{2n-2r-i-j}{n-2r-1}\right]_{i,j=1}^{r-1}$.
• The number of non-intersecting path systems from $P_i = (i,r+1)$ to $Q_i' = (n-r,n-i+1)$ equals the multiplicity of the Pfaffian ring, after removing redundant path segments.
• The method establishes a bijection between facets of the associated simplicial complex and non-intersecting path systems, enabling the use of determinant formulas from path enumeration.