On Lattice Path Counting by Major Index and Descents

A Determinantal Formula for the Hilbert Series of One-sided Ladder Determinantal Rings

“…The correspondence that we are going to describe is gleaned from [18], see also [15,Sec. 13.4] and [16].…”

“…(The concept of two-rowed arrays was introduced in [12,18] and developed to full power in [13,14]. Also the proof of the main theorem in [17] depended heavily on two-rowed arrays.…”

A remarkable formula for counting nonintersecting lattice paths in a ladder with respect to turns

“…For the proof of Theorem 1 we rely on the encoding of lattice paths in terms of two-rowed arrays, which was introduced and used in [11], [13], [14], [15], [16]. Obviously, given the starting and the end point of a path, the NE-turns uniquely determine the path.…”

“…Identity (6.6) is the special case q = 1, µ i = α i , λ i = η i , c → ∞, d = D, of [16,Theorem 1]. Equation (6.9) is also true with P LD instead of P. Application of (6.6) in the right-hand side of (6.9), with P replaced by P LD , then leads to (6.7).…”

“…The crucial idea in the proof of the conjecture is to encode lattice paths by two-rowed arrays, and then play with these two-rowed arrays. This idea was introduced in [11], [16] and developed to full power in [13], [14], [15]. The implication of Theorem 1 to the computation of Hilbert series of ladder determinantal rings is developed in section 3, see Theorem 2.…”