Labeled Partitions and the q-Derangement Numbers
نویسندگان
چکیده
Inspired by MacMahon’s original proof of his celebrated theorem on the distribution of the major index over permutations, we give a reformulation of his argument in terms of labeled partitions. In this framework, we establish a decomposition theorem for labeled partitions which is analogous to the decomposition of a permutation into derangement points and fixed points. This decomposition implies a reformulation of Wachs’ formula concerning the derangement parts and major index on permutations which was derived in order to present a bijective treatment of Gessel’s formula on the q-derangement numbers.
منابع مشابه
Labeled partitions with colored permutations
In this paper, we extend the notion of labeled partitions with ordinary permutations to colored permutations in the sense that the colors are endowed with a cyclic structure. We use labeled partitions with colored permutations to derive the generating function of the fmajk indices of colored permutations. The second result is a combinatorial treatment of a relation on the q-derangement numbers ...
متن کاملThe limiting distribution of the q-derangement numbers
We prove the normality of the limiting distribution of the coefficients of the q-derangement numbers of type B based on the formula of Foata and Han that contains a parameter z. Setting the parameter z to zero, we are led to the case of ordinary q-derangement numbers. For z = 1, we obtain the normality of the distribution of the coefficients of the usual q-derangement numbers of type B.
متن کاملThe ratio monotonicity of the q-derangement numbers
We show that the q-derangement numbers satisfy a ratio monotone property, which is analogous to the log-concavity and is stronger than the spiral property and the unimodality.
متن کاملSet partition statistics and q-Fibonacci numbers
We consider the set partition statistics ls and rb introduced by Wachs and White and investigate their distribution over set partitions avoiding certain patterns. In particular, we consider those set partitions avoiding the pattern 13/2, Πn(13/2), and those avoiding both 13/2 and 123, Πn(13/2, 123). We show that the distribution over Πn(13/2) enumerates certain integer partitions, and the distr...
متن کاملModified degenerate Carlitz's $q$-bernoulli polynomials and numbers with weight ($alpha ,beta $)
The main goal of the present paper is to construct some families of the Carlitz's $q$-Bernoulli polynomials and numbers. We firstly introduce the modified Carlitz's $q$-Bernoulli polynomials and numbers with weight ($_{p}$. We then define the modified degenerate Carlitz's $q$-Bernoulli polynomials and numbers with weight ($alpha ,beta $) and obtain some recurrence relations and other identities...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- SIAM J. Discrete Math.
دوره 22 شماره
صفحات -
تاریخ انتشار 2008