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Abstract We generalize the shuffle theorem and its $(km,kn)$ version, as conjectured by Haglund et al. and Bergeron et al. and proven by Carlsson and Mellit, and Mellit, respectively. In our version the $(km,kn)$ Dyck paths on the combinatorial side are replaced by lattice paths lying under a line segment whose x and y intercepts need not be integers, and the algebraic side is given either by a Schiffmann algebra operator formula or an equivalent explicit raising operator formula. We derive our combinatorial identity as the polynomial truncation of an identity of infinite series of $\operatorname {\mathrm {GL}}_{l}$ characters, expressed in terms of infinite series versions of LLT polynomials. The series identity in question follows from a Cauchy identity for nonsymmetric Hall–Littlewood polynomials. 

Abstract We prove the extended delta conjecture of Haglund, Remmel and Wilson, a combinatorial formula for $\Delta _{h_l}\Delta ' _{e_k} e_{n}$ , where $\Delta ' _{e_k}$ and $\Delta _{h_l}$ are Macdonald eigenoperators and $e_n$ is an elementary symmetric function. We actually prove a stronger identity of infinite series of $\operatorname {\mathrm {GL}}_m$ characters expressed in terms of LLT series. This is achieved through new results in the theory of the Schiffmann algebra and its action on the algebra of symmetric functions. 

Soil temperature and soil moisture have been measured at multiple locations at the Hubbard Brook Experimental Forest (HBEF), as part of a study of the relationships between snow depth, soil freezing and nutrient cycling (http://www.ecostudies.org/people_sci_groffman_snow_summary.html). In October 2010, we established 6, 20 x 20-m plots (intensive plots) and 14 10 x 10-m plots (extensive plots) along an elevation gradient, with eight of the plots on north-facing slopes and twelve on south-facing slopes. Soil temperature and soil moisture were measured at hourly intervals on these plots beginning in November 2010. Six locations were discontinued in September 2012 (E04, E05, E06, E11-B, E13, and E14). Previous versions of this dataset included both temperature and moisture. These data are now available as temperature (this dataset) and moisture (https://portal.edirepository.org/nis/mapbrowse?scope=knb-lter-hbr&identifier=137). These data were gathered as part of the Hubbard Brook Ecosystem Study (HBES). The HBES is a collaborative effort at the Hubbard Brook Experimental Forest, which is operated and maintained by the USDA Forest Service, Northern Research Station. 

Soil temperature and soil moisture have been measured at multiple locations at the Hubbard Brook Experimental Forest (HBEF), as part of a study of the relationships between snow depth, soil freezing and nutrient cycling (http://www.ecostudies.org/people_sci_groffman_snow_summary.html). In October 2010, we established 6, 20 x 20-m plots (intensive plots) and 14 10 x 10-m plots (extensive plots) along an elevation gradient, with eight of the plots on north-facing slopes and twelve on south-facing slopes. Soil temperature and soil moisture were measured at hourly intervals on these plots beginning in November 2010. Six locations were discontinued in September 2012 (E04, E05, E06, E11-B, E13, and E14). Previous versions of this dataset included both temperature and moisture. These data are now available as moisture(this dataset) and temperature (https://portal.edirepository.org/nis/mapbrowse?scope=knb-lter-hbr&identifier=315]. These data were gathered as part of the Hubbard Brook Ecosystem Study (HBES). The HBES is a collaborative effort at the Hubbard Brook Experimental Forest, which is operated and maintained by the USDA Forest Service, Northern Research Station. 

These data are from four separate projects undertaken between 1997 and 2017. The first of these are two snow manipulation (freeze) projects: 1) In 1997, as part of a study of the relationships between snow depth, soil freezing and nutrient cycling, we established eight 10 x 10-m plots located within four stands; two dominated (80%) by sugar maple (SM1 and SM2) and two dominated by yellow birch(YB1 and YB2), with one snow reduction (shoveling) and one reference plot in each stand. 2) In 2001, we established eight new 10-m x 10-m plots (4 treatment, 4 reference) in four new sites; two high elevation, north facing and (East Kineo and West Kineo) two low elevation, south facing (Upper Valley and Lower Valley) maple-beech-birch stands. To establish plots, we cleared minor amounts of understory vegetation from all (both treatment and reference) plots (to facilitate shoveling). Treatments (keeping plots snow free by shoveling through the end of January) were applied in the winters of 1997/98, 1998/99, 2002/2003 and 2003/2004. The Climate Gradient Project was established in October 2010. Here we evaluated relationships between snow depth, soil freezing and nutrient cycling along an elevation/aspect gradient that created variation in climate with little variation in soils or vegetation. We established 6 20 x 20-m plots (intensive plots) and 14 10 x 10-m plots (extensive plots), with eight of the plots facing north and twelve facing south. The Ice Storm project was designed to evaluate the damage and changes ice storms cause to northern hardwood forests in forest structure, nutrient cycling and carbon storage. Ten 20x30 meter plots were established in a predominately sugar maple stand, with 4 icing treatments and 2 control plots. The treatments are as follows: Low (0.25"), Mid (0.5"), Midx2 (0.5") 2 Years in a row, High: (0.75"), Control. The icing treatment was conducted in the winter of 2015-2016, with a second year of icing on the Midx2 treatments plots in the winter of 2016-2017. The treatments are as follows: Low (0.25"), Mid (0.5"), Midx2 (0.5") 2 Years in a row, High: (0.75"), Control. 

We introduce a type $A$ crystal structure on decreasing factorizations of fully-commu\-tative elements in the 0-Hecke monoid which we call $\star$-crystal. This crystal is a $K$-theoretic generalization of the crystal on decreasing factorizations in the symmetric group of the first and last author. We prove that under the residue map the $\star$-crystal intertwines with the crystal on set-valued tableaux recently introduced by Monical, Pechenik and Scrimshaw. We also define a new insertion from decreasing factorization to pairs of semistandard Young tableaux and prove several properties, such as its relation to the Hecke insertion and the uncrowding algorithm. The new insertion also intertwines with the crystal operators. 

We introduce a type A crystal structure on decreasing factorizations on 321-avoiding elements in the 0-Hecke monoid which we call *-crystal. This crystal is a K-theoretic generalization of the crystal on decreasing factorizations in the symmetric group of the first and last author. We prove that under the residue map the *-crystal intertwines with the crystal on set-valued tableaux recently introduced by Monical, Pechenik and Scrimshaw. We also define a new insertion from decreasing factorization to pairs of semistandard Young tableaux and prove several properties, such as its relation to the Hecke insertion and the uncrowding algorithm. The new insertion also intertwines with the crystal operators.