Explaining unforeseen congruence relationships between PEND and POND partitions via an Atkin-Lehner involution.

For the past several years, numerous authors have studied POD and PED partitions from a variety of perspectives. These are integer partitions wherein the odd parts must be distinct (in the case of POD partitions) or the even parts must be distinct (in the case of PED partitions). More recently, Ballantine and Welch were led to consider POND and PEND partitions, which are integer partitions wherein the odd parts be distinct (in the case of POND partitions) or the even parts be distinct (in the case of PEND partitions).

Targeting editing of tomato cis-regulatory regions generates plants with altered stomatal density in response to changing climate conditions.

Flexible developmental programs enable plants to customize their organ size and cellular composition. In leaves of eudicots, the stomatal lineage produces two essential cell types, stomata and pavement cells, but the total numbers and ratio of these cell types can vary. Central to this flexibility is the stomatal lineage initiating transcription factor, SPEECHLESS (SPCH).

A single-variable proof of the omega SPT congruence family over powers of 5.

In 2018, Liuquan Wang and Yifan Yang proved the existence of an infinite family of congruences for the smallest parts function corresponding to the third-order mock theta function . Their proof took the form of an induction requiring 20 initial relations, and utilized a space of modular functions isomorphic to a free rank 2 -module. This proof strategy was originally developed by Paule and Radu to study families of congruences associated with modular curves of genus 1.

Evolution of polarity protein BASL and the capacity for stomatal lineage asymmetric divisions.

Asymmetric and oriented stem cell divisions enable the continued production of patterned tissues. The molecules that guide these divisions include several "polarity proteins" that are localized to discrete plasma membrane domains, are differentially inherited during asymmetric divisions, and whose scaffolding activities can guide division plane orientation and subsequent cell fates. In the stomatal lineages on the surfaces of plant leaves, asymmetric and oriented divisions create distinct cell types in physiologically optimized patterns.

Single-cell resolution of lineage trajectories in the Arabidopsis stomatal lineage and developing leaf.

Dynamic cell identities underlie flexible developmental programs. The stomatal lineage in the Arabidopsis leaf epidermis features asynchronous and indeterminate divisions that can be modulated by environmental cues. The products of the lineage, stomatal guard cells and pavement cells, regulate plant-atmosphere exchanges, and the epidermis as a whole influences overall leaf growth.