This paper attempts to define a generalisation of the standard Einstein condition (in conformal/metric geometry) to any parabolic geometry. To do so, it shows that any preserved involution σ of the adjoint bundle A gives rise, given certain algebraic conditions, to a unique preferred affine connection ∇ with covariantly constant rho-tensor P, compatible with the algebraic bracket on A. These co...

We provide a combinatorial approach to the largest power of p in the number of permutations π with π = 1, for a fixed prime number p. With this approach, we find the largest power of 2 in the number of involutions, in the signed sum of involutions and in the numbers of even or odd involutions.

We complete the Wilf classification of signed patterns of length 5 for both signed permutations and signed involutions. New general equivalences of patterns are given which prove Jaggard’s conjectures concerning involutions in the symmetric group avoiding certain patterns of length 5 and 6. In this way, we also complete the Wilf classification of S5, S6, and S7 for both permutations and involut...

Let Σg,b denote a closed oriented surface genus g with b punctures and let Modg,b denote its mapping class group. In [10] Luo proved that if the genus is at least 3, the group Modg,b is generated by involutions. He also asked if there exists a universal upper bound, independent of genus and the number of punctures, for the number of torsion elements/involutions needed to generate Modg,b. Brendl...

Based on the second author’s thesis [Hor08] in this article we provide a uniform treatment of abstract involutions of algebraic groups and of Kac–Moody groups using twin buildings, RGD systems, and twisted involutions of Coxeter groups. Notably we simultaneously generalize the double coset decompositions established in [Spr84], [HW93] for algebraic groups and in [KW92] for certain Kac–Moody gro...

We consider the enumeration of pattern-avoiding involutions, focusing in particular on sets defined by avoiding a single pattern of length 4. We directly enumerate the involutions avoiding 1342 and the involutions avoiding 2341. As we demonstrate, the numerical data for these problems exhibits some surprising behavior. This strange behavior even provides some very unexpected data related to the...

We complete the Wilf classification of signed patterns of length 5 for both signed permutations and signed involutions. New general equivalences of patterns are given which prove Jaggard’s conjectures concerning involutions in the symmetric group avoiding certain patterns of length 5 and 6. In this way, we also complete the Wilf classification of S5, S6, and S7 for involutions.

We show that for k ≥ 5 and the permutations τk = (k − 1)k(k − 2) . . . 312 and Jk = k(k − 1) . . . 21, the generating tree for involutions avoiding the pattern τk is isomorphic to the generating tree for involutions avoiding the pattern Jk. This implies a family of Wilf equivalences for pattern avoidance by involutions; at least the first member of this family cannot follow from any type of pre...

We study generating functions for the number of involutions, even involutions, and odd involutions in Sn subject to two restrictions. One restriction is that the involution avoid 3412 or contain 3412 exactly once. The other restriction is that the involution avoid another pattern τ or contain τ exactly once. In many cases we express these generating functions in terms of Chebyshev polynomials o...