Quantum $\mathfrak{g}\mathfrak{l}$-weight system and its average values

We present a proof of a recent conjecture due to M. Kazarian, E. Krasilnikov, S. Lando, and M. Shapiro, which describes the average value of the universal $\mathfrak{g}\mathfrak{l}$-weight system on permutations. The proof uses a quantum analogue of the $\mathfrak{g}\mathfrak{l}$-weight system on Hecke algebras of type $A$, which leads to a one-parameter deformation of the average value of the universal $\mathfrak{g}\mathfrak{l}$-weight system. We show that the average value of the quantum weight system is a linear combination of one-part Schur functions, with coefficients being $q$-analogues of Bernoulli polynomials.