Refining the bijections among ascent sequences, (2+2)-free posets, integer matrices and pattern-avoiding permutations

The combined work of Bousquet-M\'elou, Claesson, Dukes, Jel\'inek, Kitaev, Kubitzke and Parviainen has resulted in non-trivial bijections among ascent sequences, (2+2)-free posets, upper-triangular integer matrices, and pattern-avoiding permutations. To probe the finer behavior of these bijections, we study two types of restrictions on ascent sequences. These restrictions are motivated by our results that their images under the bijections are natural and combinatorially significant. In addition, for one restriction, we are able to determine the effect of poset duality on the corresponding ascent sequences, matrices and permutations, thereby answering a question of the first author and Parviainen in this case. The second restriction should appeal to Catalaniacs.

[1]  Vít Jelínek,et al.  Counting general and self-dual interval orders , 2011, J. Comb. Theory, Ser. A.

[2]  Lara Pudwell,et al.  Enumeration Schemes for Permutations Avoiding Barred Patterns , 2010, Electron. J. Comb..

[3]  Dandan Chen,et al.  Equidistributed Statistics on Fishburn Matrices and Permutations , 2018, Electron. J. Comb..

[4]  Sergey Kitaev,et al.  (2+2)-free Posets, Ascent Sequences and Pattern Avoiding Permutations , 2008, J. Comb. Theory, Ser. A.

[5]  Filippo Disanto,et al.  Catalan pairs: A relational-theoretic approach to Catalan numbers , 2010, Adv. Appl. Math..

[6]  Vít Jelínek,et al.  Composition Matrices, (2+2)-Free Posets and their Specializations , 2011, Electron. J. Comb..

[7]  Julian West,et al.  Generating trees and the Catalan and Schröder numbers , 1995, Discret. Math..

[8]  J. Remmel,et al.  Enumerating (2+2)-free posets by indistinguishable elements , 2010, 1006.2696.

[9]  Anders Claesson,et al.  Partition and composition matrices , 2011, J. Comb. Theory, Ser. A.

[10]  George E. Andrews,et al.  On q-series identities related to interval orders , 2013, Eur. J. Comb..

[11]  Vít Jelínek,et al.  Catalan pairs and Fishburn triples , 2015, Adv. Appl. Math..

[12]  Richard P. Stanley,et al.  On Immanants of Jacobi-Trudi Matrices and Permutations with Restricted Position , 1993, J. Comb. Theory, Ser. A.

[13]  Lara K. Pudwell.  ENUMERATION SCHEMES FOR PATTERN-AVOIDING WORDS AND PERMUTATIONS , 2008 .

[14]  Einar Steingrímsson,et al.  Pattern Avoidance in Ascent Sequences , 2011, Electron. J. Comb..

[15]  George Hutchinson,et al.  Partioning algorithms for finite sets , 1963, CACM.

[16]  Mark Dukes,et al.  Ascent Sequences and Upper Triangular Matrices Containing Non-Negative Integers , 2009, Electron. J. Comb..

[17]  Gary Gordon.  Series-parallel posets and the Tutte polynomial , 1996, Discret. Math..

[18]  Jeffrey B. Remmel,et al.  Enumerating (2+2)-free posets by the number of minimal elements and other statistics , 2011, Discret. Appl. Math..

[19]  Mathieu Guay-Paquet,et al.  A modular relation for the chromatic symmetric functions of (3+1)-free posets , 2013, 1306.2400.

[20]  Stephen J. Young,et al.  Hereditary Semiorders and Enumeration of Semiorders by Dimension , 2020, Electron. J. Comb..

[21]  S. Rinaldi,et al.  Catalan Lattices on Series Parallel Interval Orders , 2010, 1007.1911.

[22]  Peter C. Fishburn,et al.  Interval orders and interval graphs : a study of partially ordered sets , 1985 .

[23]  Josef Hofbauer,et al.  Communicated by the Managing Editors , 1972 .

[24]  Kenneth P. Bogart.  An obvious proof of Fishburn's interval order theorem , 1993, Discret. Math..

[25]  Mark Dukes.  Generalized ballot sequences are ascent sequences , 2016, Australas. J Comb..