Top-degree components of Grothendieck and Lascoux polynomials

The Castelnuovo–Mumford polynomial 𝔊 ^ w with w∈S n is the highest homogeneous component of the Grothendieck polynomial 𝔊 w . Pechenik, Speyer and Weigandt define a statistic rajcode(·) on S n that gives the leading monomial of 𝔊 ^ w . We introduce a statistic rajcode(·) 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, rajcode(D) agrees with the aforementioned rajcode(w). When D is the key diagram of a weak composition α, rajcode(D) yields the leading monomial of 𝔏 ^ α , the highest homogeneous component of the Lascoux polynomials 𝔏 α . We use 𝔏 ^ α to construct a basis of V ^ n , the span of 𝔊 ^ w with w∈S n . Then we show V ^ n gives a natural algebraic interpretation of a classical q-analogue of Bell numbers.