Top-degree components of Grothendieck and Lascoux polynomials

The Castelnuovo-Mumford polynomial ${\widehat{\mathfrak{G}}}_w$ with $w\in S_n$ is the highest homogeneous component of the Grothendieck polynomial ${\mathfrak{G}}_w$. Pechenik, Speyer and Weigandt define a statistic $\mathsf{r}\mathsf{a}\mathsf{j}\mathsf{c}\mathsf{o}\mathsf{d}\mathsf{e}(\cdot )$ on $S_n$ that gives the leading monomial of ${\widehat{\mathfrak{G}}}_w$. We introduce a statistic $\mathsf{r}\mathsf{a}\mathsf{j}\mathsf{c}\mathsf{o}\mathsf{d}\mathsf{e}(\cdot )$ on any diagram $D$ through a combinatorial construction ``snow diagram’’ that augments and decorates $D$. When $D$ is the Rothe diagram of a permutation $w$, $\mathsf{r}\mathsf{a}\mathsf{j}\mathsf{c}\mathsf{o}\mathsf{d}\mathsf{e}(D)$ agrees with the aforementioned $\mathsf{r}\mathsf{a}\mathsf{j}\mathsf{c}\mathsf{o}\mathsf{d}\mathsf{e}(w)$. When $D$ is the key diagram of a weak composition $\alpha$, $\mathsf{r}\mathsf{a}\mathsf{j}\mathsf{c}\mathsf{o}\mathsf{d}\mathsf{e}(D)$ yields the leading monomial of ${\widehat{\mathfrak{L}}}_{\alpha }$, the highest homogeneous component of the Lascoux polynomials ${\mathfrak{L}}_{\alpha }$. We use ${\widehat{\mathfrak{L}}}_{\alpha }$ to construct a basis of ${\widehat{V}}_n$, the span of ${\widehat{\mathfrak{G}}}_w$ with $w\in S_n$. Then we show ${\widehat{V}}_n$ gives a natural algebraic interpretation of a classical $q$-analogue of Bell numbers.