A classical result of MacMahon gives a simple product formula for the generating function of major index over the symmetric group. A similar factorial-type product formula for the generating function of major index together with sign was given by Gessel and Simion. Several extensions are given in this paper, including a recurrence formula, a specialization at roots of unity and type B analogues.

The difference between EVALP(C,D) and EVAL(C,D) is that EVALP(C,D) fixes the value of a vertex w by i. We want to prove EVALP(C,D) ≡ EVAL(C,D). It is easy to see that EVAL(C,D) ≤ EVALP(C,D). Thus, we only need to prove the other direction. First, we define the notion of a discrete unitary matrix. Definition 1. Let F ∈ C be a matrix. We say F is M-discrete unitary for some positive integer M if ...

In this correspondence we give several new families of binary sequences of period with optimal three-level autocorrelation, where 2(mod 4). These sequences are either balanced or almost balanced. Our construction is based on cyclotomy.

We study quadratic forms that can occur as trace forms qL/K of Galois field extensions L/K, under the assumption that K contains a primitive 4th root of unity. M. Epkenhans conjectured that qL/K is always a scaled Pfister form. We prove this conjecture and classify the finite groups G which admit a G-Galois extension L/K with a nonhyperbolic trace form. We also give several applications of thes...

7 Periodic Series and Functions 128 7.1 Basic Trigonometric Identities . . . . . . . . . . . . . . . . . . 128 7.2 Basic Trigonometric Integrals . . . . . . . . . . . . . . . . . . 130 7.3 Basic Properties of Complex Numbers . . . . . . . . . . . . . 131 7.4 Complex Numbers and Trigonometrical Identities . . . . . . . 133 7.5 The Exponential of a Complex Number . . . . . . . . . . . . . 133 7.6...

Abstract. We show that for any prime number l > 2 the minus class group of the field of the l-th roots of unity Qp(ζl) admits a finite free resolution of length 1 as a module over the ring Ẑ[G]/(1 + ι). Here ι denotes complex conjugation in G = Gal(Qp(ζl)/Qp) ∼= (Z/lZ)∗. Moreover, for the primes l ≤ 509 we show that the minus class group is cyclic as a module over this ring. For these primes we...

Explicit formulae for τ ′ r(L(p, q)) and τ(L(p, q)) are obtained for all L(p, q). There are three systems of invariants of Witten-type for closed oriented 3manifolds: 1. {τr(M), r ≥ 2; τ ′ r(M), r odd ≥ 3}, where τr was defined by Reshetikhin and Turaev [1], and τ ′ r was defined by Kirby-Melvin [2] 2. {Θr(M,A), r ≥ 1,whereA is a 2r-th primitive root of unity} defined by Blanchet, Habegger, Mas...

We replace the group of group-like elements of the quantized enveloping algebra Uq(g) of a finite dimensional semisimple Lie algebra g by some regular monoid and get the weak Hopf algebra w q (g). It is a new subclass of weak Hopf algebras but not Hopf algebras. Then we devote to constructing a basis of w q (g) and determine the group of weak Hopf algebra automorphisms of w q (g) when q is not ...