Dirichlet Series, Generalized Euler And Class Numbers And Labeled Ballot Paths

The contribution of this thesis is several enumeration results of the generalized Euler and class numbers sm,n, including the generating functions of sm,n, the combina-torial interpretation of sm,n, a new combinatorial interpretation of the Springer numbers S2,n in terms of the labeled ballot paths of length n, the connection between sm,n and the derivative polynomials and so forth.In Chapter1, we summarize the background knowledge on the generalized Euler and class numbers sm,n, list the well known combinatorial interpretations of sm,n for m=1,2,3,4, and present some necessary definitions and preliminaries.In Chapter2, we obtain a formula for the exponential generating function sm(x) of sm,n, where m is an arbitrary positive integer. In particular, for m＞1, say, m=bu2, where b is square-free and u＞1, we show that sm(x) can be expressed as a linear combination of the four functions w(b,t) sec(btx)(±cos((b-p)tx)±sin(ptx)), where p is a nonnegative integer not exceeding b, t|u2and w(b,t)=Kbt/u with Kb being a constant depending on b. Moreover, the Dirichlet series Lm(s) can be easily computed from the generating function formula for sm(x).In Chapter3, we show that the main ingredient in the formula for sm,n has a combi-natorial interpretation in terms of the A-alternating augmented m-signed permutations defined by Ehrenborg and Readdy. More precisely, when m is square-free, this answers a question posed by Shanks concerning a combinatorial interpretation of the numbers Sm,n.When m is not square-free, say m=bu2, the numbers K-1b sm,n can be written as a linear combination of the numbers of A-alternating augmented bt-signed permutations with integer coefficients, where t|u2.In Chapter4, we focus on the enumeration results of the generalized Euler and class numbers S2,n, which are also called the Springer numbers. We give a new com-binatorial interpretation for the Springer numbers in terms of labeled ballot paths. In fact, we introduce the inversion code of a snake of type Bn and establish a bijection between labeled ballot paths of length n and snakes of type Bn. Moreover, we obtain the bivariate generating function for the number B(n,k) of labeled ballot paths start-ing at (0,0) and ending at (n,k), which can be viewed as a refinement of the Springer numbers. Using our bijection, we find a statistic a such that the number of snakes π of type Bn with α(π)=k equals B(n,k). A labeled ballot path that eventually returns to the x-axis is called a labeled Dyck path. We also show that our bijection specializes to a bijection between labeled Dyck paths of length2n and alternating permutations on [2n]={1,2,…,2n}.In Chapter5, for arbitrary positive integer m, we obtain the connection between the generalized Euler and class numbers and the derivative polynomials Pn(y) for the tan-gent function tanx and Qn(y) for the secant function secx. In particular, for m＞1, say m=bu2, where b is square-free and u＞1, we find that sm,n can be expressed as a linear combination of the four derivative polynomials P2n (cot(pπ/2b)), P2n-1(cot(pπ/2b)), Q2n(cot(pπ/2b)) and Q2n-1(cot(pπ/2b)), where p is an odd integer not exceeding [(2b-1)/2], and the coefficients are the four functions Kbt (-b/p) csc(pπ/2b)/, Kbt (-b/p)/(?), Kbt (b/p)/(?) and Kbt (b/p) csc(pπ/2b)/(?), where t|u4n+1and Kb is a constant depending on b. This argument not only generalizes the related work of Hoffman, but also leads to another way to compute sm,n.