Naruse hook formula for linear extensions of mobile posets

TL;DR

This paper extends the Naruse hook-length formula to mobile posets, providing new $q$-analogues for counting linear extensions, which are key objects in combinatorics.

Contribution

It introduces a Naruse type hook-length formula for mobile posets, including major index and inversion index $q$-analogues, expanding enumeration tools for these posets.

Findings

Derived a $q$-analogue for the major index for mobile posets.

Established an inversion index $q$-analogue for mobile tree posets.

Connected the formula to existing determinant formulas for mobile posets.

Abstract

Linear extensions of posets are important objects in enumerative and algebraic combinatorics that are difficult to count in general. Families of posets like Young diagrams of straight shapes and $d$-complete posets have hook-length product formulas to count linear extensions, whereas families like Young diagrams of skew shapes have determinant or positive sum formulas like the Naruse hook-length formula from 2014. In 2020, Garver et. al. gave determinant formulas to count linear extensions of a family of posets called mobile posets that refine $d$-complete posets and border strip skew shapes. We give a Naruse type hook-length formula to count linear extensions of such posets by proving a major index $q$-analogue. We also a inversion index $q$-analogue of the Naruse formula for mobile tree posets.