Gröbner bases and multiplicity of determinantal and pfaffian ideals

Algebraic and Geometric Methods in Enumerative Combinatorics

“…Then the multiplicity of the Pfaffian ring R k (X) is also given by (10). [HT92,GK04] 5. (Schröder determinants and Aztec diamonds.)…”

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

“…(We remark that the definition of ladder is more general in [1], [2], [5], [10]. However, there is in effect no loss of generality since the ladders of [1], [2], [5], [10] can always be reduced to our definition by discarding superfluous 0's.)…”

“…However, there is in effect no loss of generality since the ladders of [1], [2], [5], [10] can always be reduced to our definition by discarding superfluous 0's.) Combining results of Abhyankar [1], [2] or Herzog and Trung [10] with our Theorem 1, we are able to give an explicit formula for the Hilbert series of the ladder determinantal ring R n+1 (Y ) in the case of one-sided ladders, which can be effectively used computationally. By a one-sided ladder, we mean a ladder Y where either Y 0,0 = X 0,0 or Y b,a = X b,a .…”

“…We describe two ways to establish the theorem. One is by use of work of Abhyankar [1] (see also [2]), the other is by use of work of Herzog and Trung [10].…”

“…Recent work of Abhyankar and Kulkarni [1], [2], [18], [19], Bruns, Conca, Herzog, and Trung [3], [5], [6], [10] showed that the computation of the Hilbert series for ladder determinantal rings (see section 3 for the definition) boils down to counting families of n nonintersecting lattice paths with a given total number of turns in a certain ladder-shaped region. For one-sided ladders, Conca and Herzog [6, last paragraph] conjectured a remarkable formula that reduces the problem to counting the number of all single lattice paths with a given number of turns in the ladder region.…”