Annals of Combinatorics

Schreier-Aigner, Florian

doi: 10.1007/s00026-024-00691-5pmid: 40129605

We introduce a symmetry class for higher dimensional partitions—fully complementary higher dimensional partitions (FCPs)—and prove a formula for their generating function. By studying symmetry classes of FCPs in dimension 2, we define variations of the classical symmetry classes for plane partitions. As a by-product, we obtain conjectures for three new symmetry classes of plane partitions and prove that another new symmetry class, namely quasi-transpose-complementary plane partitions, are equinumerous to symmetric plane partitions.

D’Adderio, Michele; Iraci, Alessandro

doi: 10.1007/s00026-023-00663-1pmid: N/A

Haglund et al. (Trans Am Math Soc 370(6):4029–4057, 2018) introduced their Delta conjectures, which give two different combinatorial interpretations of the symmetric function Δen-k-1′en\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Delta '_{e_{n-k-1}} e_n$$\end{document} in terms of rise-decorated or valley-decorated labelled Dyck paths. While the rise version has been recently proved (D’Adderio and Mellit in Adv Math 402:108342, 2022; Blasiak et al. in A Proof of the Extended Delta Conjecture, arXiv:2102.08815, 2021), not much is known about the valley version. In this work, we prove the Schröder case of the valley Delta conjecture, the Schröder case of its square version (Iraci and Wyngaerd in Ann Combin 25(1):195–227, 2021), and the Catalan case of its extended version (Qiu and Wilson in J Combin Theory Ser A 175:105271, 2020). Furthermore, assuming the symmetry of (a refinement of) the combinatorial side of the extended valley Delta conjecture, we deduce also the Catalan case of its square version (Iraci and Wyngaerd 2021).

Wang, Andrew Y. Z.; Xiao, Ang

doi: 10.1007/s00026-023-00684-wpmid: N/A

There has been a tremendous amount of research on the truncated theta series in the past decade. How can we understand them combinatorially? In this paper, we investigate the truncated theorems of three classical theta series of Euler and Gauss, and provide a unified combinatorial treatment. Meanwhile, we propose a possible and more direct approach to deal with these truncated theorems.

Gunawan, Emily; Pan, Jianping; Russell, Heather M.; Tenner, Bridget Eileen

doi: 10.1007/s00026-024-00689-zpmid: N/A

We define and construct the “canonical reduced word” of a boolean permutation, and show that the RSK tableaux for that permutation can be read off directly from this reduced word. We also describe those tableaux that can correspond to boolean permutations, and enumerate them. In addition, we generalize a result of Mazorchuk and Tenner, showing that the “run” statistic influences the shape of the RSK tableau of arbitrary permutations, not just of those that are boolean.

Soukup, David

doi: 10.1007/s00026-023-00668-wpmid: N/A

We prove NP-hardness results for determining whether ice quivers are mutation equivalent to quivers with given properties, specifically, determining whether an ice quiver is mutation equivalent to an ice quiver with exactly k arrows between any two of its vertices is NP-hard. Also, determining whether an ice quiver is mutation equivalent to a quiver with no edges between frozen vertices is strongly NP-hard. Finally, we present a characterization of mutation classes of ice quivers with two mutable vertices.

Hu, Hongsheng

doi: 10.1007/s00026-024-00690-6pmid: N/A

A Coxeter group admits infinite-dimensional irreducible complex representations if and only if it is not finite or affine. In this paper, we provide a construction of some of those representations for certain Coxeter groups using some topological information of the corresponding Coxeter graphs.

Ren, Qiuyu; Zhang, Shengtong

doi: 10.1007/s00026-023-00681-zpmid: N/A

We provide a much shorter proof of Defant and Kravitz’s theorem that the length of Hitomezashi loops is congruent to 4 modulo 8. Our novel idea is to consider the length module 8 for Hitomezashi paths that take an excursion in a half-plane region.

Bao, Christopher; Cerbai, Giulio; Choi, Yunseo; Gan, Katelyn; Zhang, Owen

doi: 10.1007/s00026-024-00693-3pmid: N/A

Let s be West’s stack-sorting map, and let sT\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$s_{T}$$\end{document} be the generalized stack-sorting map, where instead of being required to increase, the stack avoids subpermutations that are order-isomorphic to any permutation in the set T. In 2020, Cerbai, Claesson, and Ferrari introduced the σ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\sigma $$\end{document}-machine s∘sσ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$s \circ s_{\sigma }$$\end{document} as a generalization of West’s 2-stack-sorting-map s∘s\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$s \circ s$$\end{document}. As a further generalization, in 2021, Baril, Cerbai, Khalil, and Vajnovski introduced the (σ,τ)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(\sigma , \tau )$$\end{document}-machine s∘sσ,τ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$s \circ s_{\sigma , \tau }$$\end{document} and enumerated Sortn(σ,τ)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\textrm{Sort}_{n}(\sigma ,\tau )$$\end{document}—the number of permutations in Sn\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$S_n$$\end{document} that are mapped to the identity by the (σ,τ)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(\sigma , \tau )$$\end{document}-machine—for six pairs of length 3 permutations (σ,τ)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(\sigma , \tau )$$\end{document}. In this work, we settle a conjecture by Baril, Cerbai, Khalil, and Vajnovski on the only remaining pair of length 3 patterns (σ,τ)=(132,321)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(\sigma , \tau ) = (132, 321)$$\end{document} for which |Sortn(σ,τ)|\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$|\textrm{Sort}_{n}(\sigma , \tau )|$$\end{document} appears in the OEIS. In addition, we enumerate Sortn(123,321)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\textrm{Sort}_n(123, 321)$$\end{document}, which does not appear in the OEIS, but has a simple closed form.

Sawant, Sagar S.

doi: 10.1007/s00026-024-00702-5pmid: N/A

The B-polynomial and quasisymmetric B-function, introduced by Awan and Bernardi, extends the widely studied Tutte polynomial and Tutte symmetric function to digraphs. In this article, we address one of the fundamental questions concerning these digraph invariants, which is, the determination of the classes of digraphs uniquely characterized by them. We solve an open question originally posed by Awan and Bernardi, regarding the identification of digraphs that result from replacing every edge of a graph with a pair of opposite arcs. Further, we address the more challenging problem of reconstructing digraphs using their quasisymmetric functions. In particular, we show that the quasisymmetric B-function reconstructs partially symmetric orientations of proper caterpillars. As a consequence, we establish that all orientations of paths and asymmetric proper caterpillars can be reconstructed from their quasisymmetric B-functions. These results enhance the pool of oriented trees distinguishable through quasisymmetric functions.

Bóna, Miklós; Costin, Ovidiu; Pittel, Boris

doi: 10.1007/s00026-024-00711-4pmid: N/A

We call a pair of vertex-disjoint, induced subtrees of a rooted tree twins if they have the same counts of vertices by out-degrees. The likely maximum size of twins in a uniformly random, rooted Cayley tree of size n→∞\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$n\rightarrow \infty $$\end{document} is studied. It is shown that the expected number of twins of size exp[(2+δ)logn·loglogn]\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\exp \bigl [(2+\delta )\sqrt{\log n\cdot \log \log n}\bigr ]$$\end{document} approaches zero, while the expected number of twins of size exp[(2-δ)logn·loglogn]\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\exp [(2-\delta )\sqrt{\log n\cdot \log \log n}\bigr ]$$\end{document} approaches infinity.