The structure of three pattern classes Av(2143, 4321), Av(2143, 4312) and Av(1324, 4312) is determined using the machinery of monotone grid classes. This allows the permutations in these classes to be described in terms of simple diagrams and regular languages and, using this, the rational generating functions which enumerate these classes are determined.

We show that permutations avoiding both of the (classical) patterns 4321 and 3241 have algebraic generating function conjectured by Vladimir Kruchinin.

We present a family-non-universal extension of the Standard Model where first two families feature both quark-lepton and electroweak-flavour unification, via $SU(4) \times Sp(4)_L Sp(4)_R$ gauge group, whereas unification for third family is realised \`a la Pati-Salam. Via staggered symmetry breaking steps, this construction offers natural explanation observed hierarchical pattern fermion masse...

We show that the principal order ideal below an element w in the Bruhat order on involutions in a symmetric group is a Boolean lattice if and only if w avoids the patterns 4321, 45312 and 456123. Similar criteria for signed permutations are also stated. Involutions with this property are enumerated with respect to natural statistics. In this context, a bijective correspondence with certain Motz...

We show that the principal order ideal of an element w in the Bruhat order on involutions in a symmetric group is a Boolean lattice if and only if w avoids the patterns 4321, 45312 and 456123. Similar criteria for signed permutations are also stated. Involutions with this property are enumerated with respect to natural statistics. In this context, a bijective correspondence with certain Motzkin...

We enumerate permutations in the two permutation classes $\text{Av}_n(312, 4321)$ and $\text{Av}_n(321, 4123)$ by number of cycles each admits. also refine this enumeration with respect to several statistics.

As part of both the Early Release Observations from the Hubble Space Telescope and the Key ProjeCt on the Extragalactic Distance Scale, we have obtained multiwavelength BVR WFPC2 images for the face-on Virgo cluster spiral galaxy M100 = NGC 4321. We report here preliminary results from those observations, in the form of a color-magnitude diagram for -11,500 stars down to V 27 mag and a luminosi...