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Abstract Define theCollatz map$${\operatorname {Col}} \colon \mathbb {N}+1 \to \mathbb {N}+1$$on the positive integers$$\mathbb {N}+1 = \{1,2,3,\dots \}$$by setting$${\operatorname {Col}}(N)$$equal to$$3N+1$$whenNis odd and$$N/2$$whenNis even, and let$${\operatorname {Col}}_{\min }(N) := \inf _{n \in \mathbb {N}} {\operatorname {Col}}^n(N)$$denote the minimal element of the Collatz orbit$$N, {\operatorname {Col}}(N), {\operatorname {Col}}^2(N), \dots $$. The infamousCollatz conjectureasserts that$${\operatorname {Col}}_{\min }(N)=1$$for all$$N \in \mathbb {N}+1$$. Previously, it was shown by Korec that for any$$\theta> \frac {\log 3}{\log 4} \approx 0.7924$$, one has$${\operatorname {Col}}_{\min }(N) \leq N^\theta $$for almost all$$N \in \mathbb {N}+1$$(in the sense of natural density). In this paper, we show that foranyfunction$$f \colon \mathbb {N}+1 \to \mathbb {R}$$with$$\lim _{N \to \infty } f(N)=+\infty $$, one has$${\operatorname {Col}}_{\min }(N) \leq f(N)$$for almost all$$N \in \mathbb {N}+1$$(in the sense of logarithmic density). Our proof proceeds by establishing a stabilisation property for a certain first passage random variable associated with the Collatz iteration (or more precisely, the closely related Syracuse iteration), which in turn follows from estimation of the characteristic function of a certain skew random walk on a$$3$$-adic cyclic group$$\mathbb {Z}/3^n\mathbb {Z}$$at high frequencies. This estimation is achieved by studying how a certain two-dimensional renewal process interacts with a union of triangles associated to a given frequency.
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This content will become publicly available on November 7, 2025
Two-sided permutation statistics via symmetric functions
Abstract Given a permutation statistic$$\operatorname {\mathrm {st}}$$, define its inverse statistic$$\operatorname {\mathrm {ist}}$$by. We give a general approach, based on the theory of symmetric functions, for finding the joint distribution of$$\operatorname {\mathrm {st}}_{1}$$and$$\operatorname {\mathrm {ist}}_{2}$$whenever$$\operatorname {\mathrm {st}}_{1}$$and$$\operatorname {\mathrm {st}}_{2}$$are descent statistics: permutation statistics that depend only on the descent composition. We apply this method to a number of descent statistics, including the descent number, the peak number, the left peak number, the number of up-down runs and the major index. Perhaps surprisingly, in many cases the polynomial giving the joint distribution of$$\operatorname {\mathrm {st}}_{1}$$and$$\operatorname {\mathrm {ist}}_{2}$$can be expressed as a simple sum involving products of the polynomials giving the (individual) distributions of$$\operatorname {\mathrm {st}}_{1}$$and$$\operatorname {\mathrm {st}}_{2}$$. Our work leads to a rederivation of Stanley’s generating function for doubly alternating permutations, as well as several conjectures concerning real-rootedness and$$\gamma $$-positivity.
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- Award ID(s):
- 2316181
- PAR ID:
- 10627481
- Publisher / Repository:
- Cambridge University Press
- Date Published:
- Journal Name:
- Forum of Mathematics, Sigma
- Volume:
- 12
- ISSN:
- 2050-5094
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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