Bijective proofs for Eulerian numbers in types B and D

Let $\left\langle{n\atop k}\right\rangle$, $\left\langle{B_n\atop k}\right\rangle$, and $\left\langle{D_n\atop k}\right\rangle$ be the Eulerian numbers in the types $A$, $B$, and $D$, respectively -- that is, the number of permutations of $n$ elements with $k$ descents, the number of signed permutations (of $n$ elements) with $k$ type $B$ descents, the number of even signed permutations (of $n$ elements) with $k$ type $D$ descents. Let $S_n(t) = \sum_{k = 0}^{n-1} \left\langle{n\atop k}\right\rangle t^k$, $B_n(t) = \sum_{k = 0}^{n}\left\langle{B_n\atop k}\right\rangle t^k$, and $D_n(t) = \sum_{k = 0}^{n}\left\langle{D_n\atop k}\right\rangle t^k$. We give bijective proofs of the identity $$B_n(t^2) = (1 + t)^{n+1}S_n(t) - 2ntS_n(t^2)$$ and of Stembridge's identity $$D_n(t) = B_n(t) - n2^{n-1}tS_{n-1}(t)\ .$$ These bijective proofs rely on a representation of signed permutations as paths. Using this representation we also establish a bijective correspondence between even signed permutations and pairs $(w, E)$ with $([n], E)$ a threshold graph and $w$ a degree ordering of $([n], E)$, which we use to obtain bijective proofs of enumerative results for threshold graphs.Comment: ALgebras, Graphs and Ordered Sets - August 26th to 28th 2020, Aug 2020, Nancy, Franc

Proceedings of the 1st International Conference on Algebras, Graphs and Ordered Sets (ALGOS 2020)

International audienceOriginating in arithmetics and logic, the theory of ordered sets is now a field of combinatorics that is intimately linked to graph theory, universal algebra and multiple-valued logic, and that has a wide range of classical applications such as formal calculus, classification, decision aid and social choice.This international conference “Algebras, graphs and ordered set” (ALGOS) brings together specialists in the theory of graphs, relational structures and ordered sets, topics that are omnipresent in artificial intelligence and in knowledge discovery, and with concrete applications in biomedical sciences, security, social networks and e-learning systems. One of the goals of this event is to provide a common ground for mathematicians and computer scientists to meet, to present their latest results, and to discuss original applications in related scientific fields. On this basis, we hope for fruitful exchanges that can motivate multidisciplinary projects.The first edition of ALgebras, Graphs and Ordered Sets (ALGOS 2020) has a particular motivation, namely, an opportunity to honour Maurice Pouzet on his 75th birthday! For this reason, we have particularly welcomed submissions in areas related to Maurice’s many scientific interests:• Lattices and ordered sets• Combinatorics and graph theory• Set theory and theory of relations• Universal algebra and multiple valued logic• Applications: formal calculus, knowledge discovery, biomedical sciences, decision aid and social choice, security, social networks, web semantics..

LIPIcs, Volume 244, ESA 2022, Complete Volume

LIPIcs, Volume 244, ESA 2022, Complete Volum