Forcing quasirandomness with 4-point permutations

TL;DR

The paper proves that any quasirandom-forcing set of $4$-point permutations must have at least five elements, advancing beyond prior results by addressing linear dependencies among gradients with a Hessian-based extension of the Implicit Function Theorem. The authors develop a general framework that combines permuton perturbations, gradient polynomials, and Hessian sign conditions to certify non-forcing, and they apply it to classify most gradient-dependent quadruples. They explicitly treat two exceptional quadruples by constructing parametrized permutons from higher-point permutations to exhibit densities equal to the random-value target $1/24$, thereby ruling out forcing in those cases. The work situates these findings within the broader program of identifying minimal quasirandom-forcing sets and motivates further exploration of forcing sets across different sizes and permutation patterns, suggesting the true minimal size may be eight for the $4$-point case.

Abstract

A combinatorial object is said to be quasirandom if it exhibits certain properties that are typically seen in a truly random object of the same kind. It is known that a permutation is quasirandom if and only if the pattern density of each of the twenty-four 4-point permutations is close to 1/24, which is its expected value in a random permutation. In other words, the set of all twenty-four 4-point permutations is quasirandom-forcing. Moreover, it is known that there exist sets of eight 4-point permutations that are also quasirandom-forcing. Breaking the barrier of linear dependency of perturbation gradients, we show that every quasirandom-forcing set of 4-point permutations must have cardinality at least five.

Figures (8)

• Figure 1: Visualization of the permutation $15234$.
• Figure 2: Visualization of the twelve quadruples listed in Proposition \ref{['prop:one']}. Each cell of the grid contains the index of the permutation with its support in that cell.
• Figure 3: Visualization of the seven quadruples listed in Proposition \ref{['prop:zero']}. Each cell of the grid contains the indices of the permutations with their support in that cell.
• Figure 4: Visualization of the permuton $\mu_{x,y,z}$ defined in Section \ref{['sec:special1']}.
• Figure 5: A plot of the curves $(g_1(s,t),g_2(s,t))$ for $(s,t)=(s,0)$, $(s,t)=(s,1/4)$, $(s,t)=(s,1/2)$ and $(s,t)=(s,1)$. In addition, the curve for $(s,t)=(1,t)$ is drawn as dashed.

Theorems & Definitions (42)

• Theorem 1
• Lemma 2
• proof
• Theorem 3: Implicit Function Theorem
• Corollary 4
• Theorem 5
• proof
• Corollary 6
• proof
• Lemma 7: Kurečka Kur22