Projective bundles.
Projective planes of order 10 do not have a collineation of order 3.
Projective representations of generalized symmetric groups.
Promotion and evacuation on standard Young tableaux of rectangle and staircase shape.
Promotion operator on rigged configurations of type .
Proof of a conjecture of Chan, Robbins, and Yuen.
Proof of a conjecture of Hirschhorn and Sellers on overpartitions
Let p̅(n) denote the number of overpartitions of n. It was conjectured by Hirschhorn and Sellers that p̅(40n+35) ≡ 0 (mod 40) for n ≥ 0. Employing 2-dissection formulas of theta functions due to Ramanujan, and Hirschhorn and Sellers, we obtain a generating function for p̅(40n+35) modulo 5. Using the (p, k)-parametrization of theta functions given by Alaca, Alaca and Williams, we prove the congruence p̅(40n+35) ≡ 0 (mod 5) for n ≥ 0. Combining this congruence and the congruence p̅(4n+3) ≡ 0 (mod...
Proof of a conjectured three-valued family of Weil sums of binomials
We consider Weil sums of binomials of the form , where F is a finite field, ψ: F → ℂ is the canonical additive character, , and . If we fix F and d, and examine the values of as a runs through , we always obtain at least three distinct values unless d is degenerate (a power of the characteristic of F modulo ). Choices of F and d for which we obtain only three values are quite rare and desirable in a wide variety of applications. We show that if F is a field of order 3ⁿ with n odd, and with...
Proof of a determinant evaluation conjectured by Bombieri, Hunt and van Poorten.
Proof of an intersection theorem via graph homomorphisms.
Proof of Conway's lost cosmological theorem.
Proof of Simoes-Pereira's conjectures on locally k-degenerate graphs.
Proof of the alternating sign matrix conjecture.
Proof of the Macdonald Conjecture.
Proof of the conjecture for large .
Proof of the Razumov-Stroganov conjecture for some infinite families of link patterns.
Proof of the refined alternating sign matrix conjecture.
Proof trees for weak achievement games.
Propagation of mean degrees.
Proper connection number of bipartite graphs
An edge-colored graph is proper connected if every pair of vertices is connected by a proper path. The proper connection number of a connected graph , denoted by , is the smallest number of colors that are needed to color the edges of in order to make it proper connected. In this paper, we obtain the sharp upper bound for of a general bipartite graph and a series of extremal graphs. Additionally, we give a proper -coloring for a connected bipartite graph having and a dominating cycle...