Combinatorics of hexagonal fully packed loop configurations

This paper introduces fully packed loop configurations of hexagonal shape (HFPLs) as a generalization of triangular fully packed loop configurations. To encode the boundary conditions of an HFPL, a sextuple (lT,t,rT;lB,b,rB)(lT,t,rT;lB,b,rB) of 01-words is assigned to it. The first main result of this article establishes necessary conditions for the boundary (lT,t,rT;lB,b,rB)(lT,t,rT;lB,b,rB) of an HFPL. The inequality d(rB)+d(b)+d(lB)≥d(lT)+d(t)+d(rT)+|lT|1|t|0+|t|1|rT|0+|rB|0|lB|1d(rB)+d(b)+d(lB)≥d(lT)+d(t)+d(rT)+|lT|1|t|0+|t|1|rT|0+|rB|0|lB|1 is an example of one such condition (here |⋅|i|⋅|i denotes the number of occurrences of i   and d(⋅)d(⋅) denotes the number of inversions). The other main results of this article are expressions in terms of Littlewood–Richardson coefficients for the numbers of HFPLs with boundary (lT,t,rT;lB,b,rB)(lT,t,rT;lB,b,rB) such that d(rB)+d(b)+d(lB)−d(lT)−d(t)−d(rT)−|lT|1|t|0−|t|1|rT|0−|rB|0|lB|1=0,1d(rB)+d(b)+d(lB)−d(lT)−d(t)−d(rT)−|lT|1|t|0−|t|1|rT|0−|rB|0|lB|1=0,1.