Maximal Cells in Shifted Staircase Tableaux and a Quarter-Circle Law

TL;DR

Maximal Cells in Shifted Staircase Tableaux and a Quarter-Circle Law investigates the distribution of the maximal cell in uniform standard Young tableaux of shifted staircase shape $(2n-1,2n-3,\ldots,3,1)$ and proves that the location, after scaling by $n$, converges to a density $f(x)=\frac{4}{\pi}\sqrt{1-x^2}$ on $[0,1]$, the quarter-circle law. The analysis uses Haiman's bijection between shifted tableaux and reduced words for the longest element of the type $B_n$ Coxeter group, translating combinatorial questions into reduced word statistics. The paper also provides exact formulas for the probability that a given cell carries the maximal label, and supplements this with computational experiments on random sorting networks of type $B$, including letter-frequency profiles, trajectories, and signed permutation matrix evolution. These results establish a natural type $B$ analogue to the type $A$ random sorting network and open directions for proofs of the quarter-circle limit and extensions to other Coxeter types.

Abstract

In this note, we explicitly compute the probability that a given cell in a random standard Young tableau of the shifted staircase shape $(2n-1, 2n-3, \ldots, 3,1)$ contains the maximal label. We also show that the asymptotic distribution of the cell containing the maximal label is governed by the quarter-circle law. The bijection between the tableaux and thereduced decompositions of the longest element of the group $B_n$ of the signed permutations yields the probability distribution of the first (and any) letter of the random reduced decompositions. We also show the results of some computational experiments on the random sorting networks of $B_n$.

Figures (5)

• Figure 1: The shifted shape $(4,2,1)$ and the hook $H_{1,2}$ whose length is $5$ (top left), hooklengths of cells in the Young diagram (top right), and all of the Young tableaux of shape $\lambda$
• Figure 2: Cells whose hooklengths in $\lambda$ differ from those in $\lambda'=\lambda - \{(n-r,n+r)\}$, where $r=3$
• Figure 3: The frequency of the generating elements of a reduced expression of the longest element of $B_{3000}$ chosen uniformly at random.
• Figure 4: Trajectories of particles in a random type-$B$ sorting network ($n=3000$). Each curve represents the position of one element as it evolves under the sequence of signed adjacent transpositions. The trajectories form smooth sinusoidal arcs reflected at height $0$, illustrating the characteristic symmetry of type $B$.
• Figure 5: Time evolution of the signed permutation matrix in a random type-$B$ sorting network ($n=3000$). Red and blue dots represent $+1$ and $-1$ entries, respectively. The configuration evolves smoothly from the red-dominant (left) to the blue-dominant (right) state, exhibiting quarter-circle shaped regions that reflect the symmetry of type $B$.

Theorems & Definitions (4)

• Theorem 1
• Theorem 2
• proof
• proof : Proof of Theorem\ref{['thm:main']}