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Between Weak and Bruhat: The Middle Order on Permutations.

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Title: Between Weak and Bruhat: The Middle Order on Permutations.
Authors: Bouvel, Mathilde1 (AUTHOR) mathilde.bouvel@loria.fr, Ferrari, Luca2 (AUTHOR) luca.ferrari@unifi.it, Tenner, Bridget Eileen3 (AUTHOR) bridget@math.depaul.edu
Source: Graphs & Combinatorics. Apr2025, Vol. 41 Issue 2, p1-25. 25p.
Subjects: Möbius function, Heyting algebras, Euler characteristic, Terms & phrases
Abstract: We define a partial order P n on permutations of any given size n, which is the image of a natural partial order on inversion sequences. We call this the "middle order." We demonstrate that the poset P n refines the weak order on permutations and admits the Bruhat order as a refinement, justifying the terminology. These middle orders are distributive lattices and we establish some of their combinatorial properties, including characterization and enumeration of intervals and boolean intervals (in general, or of any given rank), and a combinatorial interpretation of their Euler characteristic. We further study the (not so well-behaved) restriction of this poset to involutions, obtaining a simple formula for the Möbius function of principal order ideals there. Finally, we offer further directions of research, initiating the study of the canonical Heyting algebra associated with P n , and defining a parking function analogue of P n . [ABSTRACT FROM AUTHOR]
Copyright of Graphs & Combinatorics is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: We define a partial order P n on permutations of any given size n, which is the image of a natural partial order on inversion sequences. We call this the "middle order." We demonstrate that the poset P n refines the weak order on permutations and admits the Bruhat order as a refinement, justifying the terminology. These middle orders are distributive lattices and we establish some of their combinatorial properties, including characterization and enumeration of intervals and boolean intervals (in general, or of any given rank), and a combinatorial interpretation of their Euler characteristic. We further study the (not so well-behaved) restriction of this poset to involutions, obtaining a simple formula for the Möbius function of principal order ideals there. Finally, we offer further directions of research, initiating the study of the canonical Heyting algebra associated with P n , and defining a parking function analogue of P n . [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of Graphs & Combinatorics is the property of Springer Nature and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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