Descent generating polynomials for ($n-3$)- and ($n-4$)-stack-sortable (pattern-avoiding) permutations

In this paper, we find distribution of descents over $(n-3)$- and $(n-4)$-stack-sortable permutations in terms of Eulerian polynomials. Our results generalize the enumeration results by Claesson, Dukes, and Steingr\'{\i}msson on $(n-3)$- and $(n-4)$-stack-sortable permutations. Moreover, we find distribution of descents on $(n-2)$-, $(n-3)$- and $(n-4)$-stack-sortable permutations that avoid any given pattern of length 3, which extends known results in the literature on distribution of descents over pattern-avoiding 1- and 2-stack-sortable permutations. Our distribution results also give enumeration of $(n-2)$-, $(n-3)$- and $(n-4)$-stack-sortable permutations avoiding any pattern of length 3. One of our conjectures links our work to stack-sorting with restricted stacks, and the other conjecture states that 213-avoiding permutations sortable with $t$ stacks are equinumerous with 321-avoiding permutations sortable with $t$ stacks for any $t$.