Bounded affine permutations I. Pattern avoidance and enumeration

We introduce a new boundedness condition for affine permutations, motivated by the fruitful concept of periodic boundary conditions in statistical physics. We study pattern avoidance in bounded affine permutations. In particular, we show that if $\tau$ is one of the finite increasing oscillations, then every $\tau$-avoiding affine permutation satisfies the boundedness condition. We also explore the enumeration of pattern-avoiding affine permutations that can be decomposed into blocks, using analytic methods to relate their exact and asymptotic enumeration to that of the underlying ordinary permutations. Finally, we perform exact and asymptotic enumeration of the set of all bounded affine permutations of size $n$. A companion paper will focus on avoidance of monotone decreasing patterns in bounded affine permutations. Comment: 35 pages

Parity duality for the amplituhedron

The (tree) amplituhedron $\mathcal {A}_{n,k,m}(Z)$ is a certain subset of the Grassmannian introduced by Arkani-Hamed and Trnka in 2013 in order to study scattering amplitudes in $N=4$ supersymmetric Yang–Mills theory. Confirming a conjecture of the first author, we show that when $m$ is even, a collection of affine permutations yields a triangulation of $\mathcal {A}_{n,k,m}(Z)$ for any $Z\in \operatorname {Gr}_{>0}(k+m,n)$ if and only if the collection of their inverses yields a triangulation of $\mathcal {A}_{n,n-m-k,m}(Z)$ for any $Z\in \operatorname {Gr}_{>0}(n-k,n)$. We prove this duality using the twist map of Marsh and Scott. We also show that this map preserves the canonical differential forms associated with the corresponding positroid cells, and hence obtain a parity duality for amplituhedron differential forms.

On affine classification of permutations on the space GF(2)3

We give an elementary proof that by multiplication on left and right by affine permutations A, B ∈ AGL(3, 2) each permutation π : GF(2)3 → GF(2)3 may be reduced to one of the 4 permutations for which the 3 × 3-matrices consisting of the coefficients of quadratic terms of coordinate functions have as an invariant the rank, which is either 3, or 2, or 1, or 0, respectively. For comparison, we evaluate the number of classes of affine equivalence by the Pólya enumerative theory.

Cycles in the Graph of Overlapping Permutations Avoiding Barred Patterns

As a variation of De Bruijn graphs on strings of symbols, the graph of overlapping permutations has a directed edge $\pi(1)\pi(2)\ldots \pi(n+1)$ from the standardization of $\pi(1)\pi(2)\ldots \pi(n)$ to the standardization of $\pi(2)\pi(3)\ldots \pi(n+1)$. In this paper, we consider the enumeration of $d$-cycles in the subgraph of overlapping $(231, 4\bar{1}32)$-avoiding permutations. To this end, we introduce the notions of marked Motzkin paths and marked Riordan paths, where a marked Motzkin (resp. Riordan) path is a Motzkin (resp. Riordan) path in which exactly one step before the leftmost return point is marked. We show that the number of closed walks of length $d$ in the subgraph of overlapping $(231, 4\bar{1}32)$-avoiding permutations are closely related to the number of marked Motzkin paths and that of marked Riordan paths. By establishing bijections, we get the enumerations of marked Motzkin paths and marked Riordan paths. As a corollary, we provide bijective proofs of two identities involving Catalan numbers in answer to the problem posed by Ehrenborg, Kitaev and Steingrímsson. Moreover, we get the enumerations of $(231, 4\bar{1}32)$-avoiding affine permutations and $(312, 32\bar{4}1)$-avoiding affine permutations.