For a fixed positive integer n, let Sn denote the symmetric group of n! permutations on n symbols, and let maj(σ) denote the major index of a permutation σ. Fix positive integers k < l ≤ n, and nonnegative integers i, j. Let mn(i\k; j\l) denote the cardinality of the set {σ ∈ Sn : maj(σ) ≡ i mod k,maj(σ) ≡ j mod l}. In this paper we give some enumerative formulas for these numbers. When l divid...

where St and S2 are projective representations of G. An irreducible projective representation is one that is not reducible. In this paper, we study the degrees of irreducible projective representations of a finite group G over an algebraically closed field K. In Theorem 2, we require further that char KJ(\G\. The degree of an irreducible projective representation of a finite group G, over the c...

Lagrange’s theorem tells us that if G is a finite group and H ≤ G, then #(H) divides #(G). As we have seen, the converse to Lagrange’s theorem is false in general: if G is a finite group of order n and d divides n, then there need not exist a subgroup of G whose order is d. The Sylow theorems say that such a subgroup exists in one special but very important case: when d is the largest power of ...

A conjecture of P. Erdos says that the diophantine equation x+y = z has infinitely many solutions with (x,y) — 1 and such that if a prime p divides xyz, then p divides xyz. In this paper, we give a proof of this conjecture. Let k ^ 2 be an integer. A /c-powerful number is a positive integer x with the property that if the prime p divides x, then p divides x (2-powerful numbers were called initi...

It has been conjectured that asymptotically almost all matroids are sparse paving, i.e. that s(n) ∼ m(n), where m(n) denotes the number of matroids on a fixed groundset of size n, and s(n) the number of sparse paving matroids. In an earlier paper, we showed that log s(n) ∼ logm(n). The bounds that we used for that result were dominated by matroids of rank r ≈ n/2. In this paper we consider the ...

If n≡0,1mod4, we prove a sum formula Vθ0(a0,aRn)=n⋅Vθ0(a0,aR) for the generalized Vaserstein symbol whenever R is smooth affine algebra over perfect field k with char(k)≠2 such that −1∈k×2. This enables us to generalize result of Fasel-Rao-Swan on transformations unimodular rows via elementary matrices normal algebras dimension d≥4 algebraically closed fields characteristic ≠2. As consequence, ...