The generalized Delta conjecture at t=0

We prove the cases q=0 and t=0 of the generalized Delta conjecture of Haglund, Remmel and Wilson involving the symmetric function $\Delta_{h_m}\Delta_{e_{n-k-1}}'e_n$. Our theorem generalizes recent results by Garsia, Haglund, Remmel and Yoo. This proves also the case q=0 of our recent generalized Delta square conjecture.Comment: 21 pages, 3 figure

SERVIÇO SOCIAL, PRODUÇÃO DO ESPAÇO E A EFETIVAÇÃO DA POLÍTICA DE ASSISTÊNCIA SOCIAL

Resumo O texto orienta-se pela reflexão do Serviço Social, enquanto profissão integrada na divisão social do trabalho, nos marcos da sociedade capitalista, acerca da produção social do espaço, de modo específico na região oeste de Santa Catarina. O estudo busca desvendar elementos do atual processo de desenvolvimento capitalista do campo, compreendendo-o enquanto uma totalidade sócio-histórica, profundamente contraditória, desigual e, em movimento. Assim, a partir da análise do desenvolvimento capitalista no campo, o estudo busca identificar as expressões da questão social na contemporaneidade e os novos desafios postos ao Serviço Social, considerando especialmente a inclusão do território, enquanto eixo estruturante da Política de Assistência Social

Determinants and timing of dropping out decisions: evidence from the UK FE sector

This paper investigates whether the hazard of dropping out for both male and female students changes over the duration of study. Using duration modelling techniques wend a certain degree of non-monotonic duration dependence for both males and females. However this pattern for female students aiming at high level qualifications is sensitive to attempts to control for unobserved heterogeneity. For these students the extended models show a flattened hazard function, suggesting that the hazard is basically constant over time. For males introducing controls for unobserved heterogeneity does not change the pattern of the duration dependence, suggesting that they might be at higher risk of dropping out during the first semester of their studies. In addition, we examine variations in drop out hazard patterns for students enrolled on courses which confer different qualification levels. We provide evidence of distinct hazard patterns between students pursuing 'high level' and 'low level' qualifications

Some consequences of the valley Delta conjectures

In (Haglund, Remmel, Wilson 2018) Haglund, Remmel and Wilson introduced their Delta conjectures, which give two different combinatorial interpretations of the symmetric function $\Delta'_{e_{n-k-1}} e_n$ in terms of rise-decorated or valley-decorated labelled Dyck paths respectively. While the rise version has been recently proved (D'Adderio, Mellit 2021; Blasiak, Haiman, Morse, Pun, Seelinger preprint 2021), not much is known about the valley version. In this work we prove the Schr\"oder case of the valley Delta conjecture, the Schr\"oder case of its square version (Iraci, Vanden Wyngaerd 2021), and the Catalan case of its extended version (Qiu, Wilson 2020). Furthermore, assuming the symmetry of (a refinement of) the combinatorial side of the extended valley Delta conjecture, we deduce also the Catalan case of its square version (Iraci, Vanden Wyngaerd 2021).Comment: 19 pages, 3 figures. arXiv admin note: text overlap with arXiv:2003.1204

Delta and Theta Operator Expansions

We give an elementary symmetric function expansion for $M\Delta_{m_\gamma e_1}\Pi e_\lambda^{\ast}$ and $M\Delta_{m_\gamma e_1}\Pi s_\lambda^{\ast}$ when $t=1$ in terms of what we call $\gamma$-parking functions and lattice $\gamma$-parking functions. Here, $\Delta_F$ and $\Pi$ are certain eigenoperators of the modified Macdonald basis and $M=(1-q)(1-t)$. Our main results in turn give an elementary basis expansion at $t=1$ for symmetric functions of the form $M \Delta_{Fe_1} \Theta_{G} J$ whenever $F$ is expanded in terms of monomials, $G$ is expanded in terms of the elementary basis, and $J$ is expanded in terms of the modified elementary basis $\{\Pi e_\lambda^\ast\}_\lambda$. Even the most special cases of this general Delta and Theta operator expression are significant; we highlight a few of these special cases. We end by giving an $e$-positivity conjecture for when $t$ is not specialized, proposing that our objects can also give the elementary basis expansion in the unspecialized symmetric function.Comment: 38 pages, 12 figure

The new dinv is not so new

In [Duane, Garsia, Zabrocki 2013] the authors introduced a new dinv statistic, denoted ndinv, on the two part case of the shuffle conjecture (Haglund et al. 2005) in order to prove a compositional refinement. Though in [Hicks, Kim 2013] a non-recursive (but algorithmic) definition of ndinv has been given, this statistic still looks a bit unnatural. In this paper we "unveil the mystery" around the ndinv, by showing bijectively that the ndinv actually matches the usual dinv statistic in a special case of the generalized Delta conjecture in [Haglund, Remmel, Wilson 2018]. Moreover, we give also a non-compositional proof of the "ehh" case of the shuffle conjecture (after [Garsia, Xin, Zabrocki 2014]) by bijectively proving a relation with the two part case of the Delta conjecture