Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2609.13214

Genlex Gray codes for $S_n$ with the fewest operations: classification and symmetry

Yehonathan Sharvit
View original source ↗Read source document ↗Export metadata ↓

Abstract

A Gray code for $S_n$ is \emph{genlex} when words sharing a suffix are consecutive. We determine the genlex Gray codes for $S_n$ that use the fewest possible operations: Zaks' recursion generalises to a superfactorial family of such codes, and no code outside this family attains the minimum. Every code in the family closes into a cycle, and the cycle is invariant under a group of left translations, cyclic of order $n$ or dihedral of order $2n$ according to an explicit criterion on the operations. In the pancake graph the family reduces to a single member, the classical Zaks order; with respect to the standard parabolic chain of $W(A_{n-1})$, that order attains one extreme of total Coxeter length.

Explore related subjects

math.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Yehonathan Sharvit. 2026-08-26. Genlex Gray codes for $S_n$ with the fewest operations: classification and symmetry. https://arxiv.org/abs/2609.13214

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Orthogonal Pairs in Maps from the Sphere to the Circle

We prove that, for any $f:S^2\to S^1$ and any $\varepsilon>0$, there exist orthogonal vectors $x,y\in S^2$ such that the length of the shortest arc between $f(x)$ and $f(y)$ is at most $π/2 +\varepsilon$. This proves a conjecture of Ghebleh from 2007 that the circular chromatic number of the real orthogonality graph is equal to four.

math.CO↗
arXiv2026

Exact Area-Range Minima in the Quantitative Monsky Problem for Five and Seven Triangles

For a dissection $D$ of the unit square into $n$ nondegenerate triangles, let $R(D)=\max_i a_i-\min_i a_i, Δ(n)=\inf_D R(D).$ We prove that this infimum is attained for every $n\ge2$, and determine the exact minima for $n=5$ and $n=7$, allowing T-junctions. For five triangles, $Δ(5)=\frac{5\sqrt5-11}{8};$ equality holds precisely when three areas equal $(3-\sqrt5)/4$ and two equal $(3\sqrt5-5)/8$. For seven triangles, $Δ(7)=r_7$, where $r_7$ is the unique root in $(0,1/4900)$ of $864r^4+2160r^3-6060r^2+4972r-1.$ Every minimizer has four areas $(1+3r_7)/7$ and three areas $(1-4r_7)/7$, although its geometry need not be unique. The proofs combine finite combinatorial classification with exact symbolic and integer-interval certificates. For nine triangles, a tilted-strip construction gives the explicit algebraic upper bound $Δ(9)\le 0.0001273496861283553341\ldots,$ which is the exact minimum within that topology. Conversely, every dissection in the complete single-cap two-rail zig-zag family, with arbitrary continuous areas, has range greater than $1/3500$; hence a global minimizer must lie outside that family. The exact value of $Δ(9)$ remains open.

math.CO↗
arXiv2026

Chromatic symmetric functions for annular webs

We introduce a combinatorial definition of chromatic symmetric functions for annular webs. We prove their symmetry by constructing a web analogue of the Shareshian--Wachs involution and show that they coincide with the symmetric functions associated to annular webs via Turaev's isomorphism. We then derive explicit formulas for their hook Schur coefficients. We also introduce web LLT functions, whose hook Schur coefficients admit positive Laurent-polynomial formulas. These formulas yield a combinatorial expression for the coefficients of the HOMFLY--PT polynomial of an annular web.

math.CO↗