Record-biased permutations and their permuton limit

TL;DR

Record-biased permutations weight each permutation by $w(\sigma)=\theta^{rec(\sigma)}$, capturing notions of presortedness. The authors relate this model to the Ewens distribution via Foata's bijection, and provide multiple generative constructions and linear-time samplers. They derive explicit and asymptotic statistics for various regimes of $\theta$, and, in the linear regime $\theta=\lambda n$, prove a full permuton limit, describing it via a curve $y=f_\lambda(x)=\dfrac{x(\lambda+1)}{\lambda+x}$ with a density on the curve and a complementary under-curve density. This yields a precise limit shape for the diagram and connects to the permuton literature, offering both theoretical insights and practical sampling methods.

Abstract

In this article, we study a non-uniform distribution on permutations biased by their number of records that we call \emph{record-biased permutations}. We give several generative processes for record-biased permutations, explaining also how they can be used to devise efficient (linear) random samplers. For several classical permutation statistics, we obtain their expectation using the above generative processes, as well as their limit distributions in the regime that has a logarithmic number of records (as in the uniform case). Finally, increasing the bias to obtain a regime with an expected linear number of records, we establish the convergence of record-biased permutations to a deterministic permuton, which we fully characterize. This model was introduced in our earlier work [N. Auger, M. Bouvel, C. Nicaud, C. Pivoteau, \emph{Analysis of Algorithms for Permutations Biased by Their Number of Records}, AofA 2016], in the context of realistic analysis of algorithms. We conduct here a more thorough study but with a theoretical perspective.

Figures (15)

• Figure 1: The diagram and the set of cycles representations of $\tau =6321745$.
• Figure 2: the Chinese Restaurant process for permutations in $\mathop{\mathrm{\mathfrak{S}}}\nolimits_{3}$ in the Ewens model.
• Figure 3: Feller coupling for permutations in $\mathop{\mathrm{\mathfrak{S}}}\nolimits_{3}$ in the Ewens model.
• Figure 4: Generative process for record-biased permutations in $\mathop{\mathrm{\mathfrak{S}}}\nolimits_{3}$, viewed as sequences of sequences.
• Figure 5: Generative process for record-biased permutations in $\mathop{\mathrm{\mathfrak{S}}}\nolimits_{3}$, viewed as words.

Theorems & Definitions (54)

• Theorem 1
• proof
• Corollary 2
• proof
• Remark 3
• Theorem 4
• proof
• Corollary 5
• proof
• Remark 6