Combinatorial properties of the numbers of tableaux of bounded height

We introduce an infinite family of lower triangular matrices $\Gamma^{(s)}$, where $\gamma_{n,i}^s$ counts the standard Young tableaux on $n$ cells and with at most $s$ columns on a suitable subset of shapes. We show that the entries of these matrices satisfy a three-term row recurrence and we deduce recursive and asymptotic properties for the total number $\tau_s(n)$ of tableaux on $n$ cells and with at most $s$ columns.Comment: 11 pages, 1 figur

Continuing professional development for the early years workforce in England since 2015: a synthesis of survey data highlighting commonalities, discrepancies and gaps

Continuing Professional Development (CPD) is an essential aspect of Early Years (EY) workforce planning. In understanding how CPD is currently provided, we depend largely on surveys carried out by separate sector organisations with little synthesis of the data. In this paper, we scrutinise findings relating to CPD from nine surveys of EY managers and practitioners carried out by three well-established sector organisations since 2015. We highlight commonalities, discrepancies and gaps in the data. On the basis of this analysis, we highlight the need for further investigation around what constitutes CPD in the EY sector, what are perceived to be ‘quality’ experiences, CPD accreditation and the leadership and management of CPD in settings. We recommend gathering more detailed and accurate data about CPD through practitioner logs completed as and when CPD is undertaken, as opposed to retrospective surveys