Abstract
The lattice of monotone triangles $(\mathfrak{M}_n,\leq)$ ordered by entry-wise comparisons is studied. Let $\tau_{\min}$ denote the unique minimal element in this lattice, and $\tau_{\max}$ the unique maximum. The number of $r$-tuples of monotone triangles $(\tau_1,\ldots,\tau_r)$ with minimal infimum $\tau_{\min}$ (maximal supremum $\tau_{\max}$, resp.) is shown to asymptotically approach $r|\mathfrak{M}_n|^{r-1}$ as $n \to \infty$. Thus, with high probability this event implies that one of the $\tau_i$ is $\tau_{\min}$ ($\tau_{\max}$, resp.). Higher-order error terms are also discussed.
Highlights
Introduction and statement of resultsLet n 1 be an integer and [n] := {1, 2, . . . , n}
We let Mn denote the set of all monotone triangles of size n, and we let τ = (τ (i, j)) be a generic element from this set. It is well-known that there is a bijection between Mn and the set of all n × n alternating-sign matrices (ASMs), which are the n × n matrices of 0s, 1s, and −1s so that the sum of the entries in each row and column is 1 and the non-zero entries in each row and column alternate in sign
The set Mn is in bijection with the square ice configurations of order n in statistical mechanics; for descriptions of these bijections, along with other sets in bijection with Mn, see the surveys of James Propp [15], David Bressoud and Propp [3], or the wonderful exposition by Bressoud about the proof of the ASM conjecture [2], which concerns the number of monotone triangles, denoted A(n) := |Mn|
Summary
Let n 1 be an integer and [n] := {1, 2, . . . , n}. A monotone triangle of size n (or Gog triangle in the terminology of Doron Zeilberger [18]) is a triangular arrangement of n(n + 1)/2 integers with i elements in row i (i ∈ [n]) taken from the set [n] so that if a(i, j) denotes the jth entry in row i (counted from the top), . All entries in both monotone triangles are the same except for the first entries in rows 1 and 2, and there we have τ1(1, 1) > τ2(1, 1) and yet τ1(2, 1) < τ2(2, 1) This poset is a lattice under the infimum and supremum operations inf(τ1, τ2) := (min{τ1(i, j), τ2(i, j)}) and sup(τ1, τ2) := (max{τ1(i, j), τ2(i, j)}). It is left to the interested reader to check that these inf and sup operations do deliver monotone triangles, and satisfy the requirements of an infimum or supremum. Let τmin denote the (unique) minimal element in the monotone triangle lattice. How likely is it that r independent and uniformly random monotone triangles τ1, .
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