Unimodality Properties Of The Eulerian Distribution On The Involutions Of The Hyperoctahedral Group

Unimodality problem is one of the primary topics in combinatorics,including the study of unimodality,symmetry,log-concavity(log-convexity),γ-positivity and real-rootedness,et al.An involution is a permutation which consists only of one-cycles(fixed-points)and two-cycles(transpositions).The Eulerian distribution on the involutions of the symmetric group Sn has nice properties.In recent years,many scholars have generalized some enumerative results from the symmetric group Sn to the hyperoctahedral group Bn.In addition,they have introduced and studied several statistics for Bn,which also included some properties of the Eulerian distribution on the involutions.The paper is devoted to investigate the unimodality and γ-positivity of the Eulerian distribution on the involutions of the hyperoctahedral group.The main frame of this paper is as follows.In Chapter 1,we introduce some definitions and notions involved in this paper,and summarize the unimodality properties of the Eulerian distribution on the involutions of the hyperoctahedral group and the main work of this paper.In Chapter 2,we study the γ-positivity of InB(t),which counts the Eulerian distribution on the involutions of the hyperoctahedral group.We confirm a conjecture of Moustakas which states that InB(t)has the γ-positivity for n≥ 1 with algebraic methods.We derive the recurrence relations for the γ-expansion coefficients based on the generating function of InB(t),and prove the γ-positivity of InB(t)by mathematical induction.In Chapter 3,we discuss some unimodality properties of JnB(t),which counts the Eulerian distribution on the fixed-point free involutions of the hyperoctahedral group.We first obtain the generating function of JB(t)based on the signed quasisymmetric functions,which plays a key role in studying the unimodality properties of JnB(t).Next using the generating function of JnB(t),we derive two recurrence relations for the coefficients of JnB(t).Finally,we prove the symmetry,unimodality and γ-positivity of JnB(t)combining mathematical induction and two recurrence relations.In addition,we show that JnB(t)is not log-concave from a counterexample.In Chapter 4,we briefly summarize the content of this paper.