On page 13 of "Tilting modules for the Auslander algebra of $K(x)/x^n$" the author, Geuenich, suggests that the number ($p_{n,i}$) of isomorphism classes of modules, occurring as the $i$-th summand of some tilting module $T_x$ for the finite Auslander algebra $Λ_n$ are enumerated by the number triangle OEIS A046802.
The analysis in the paper involves the symmetric groups $S_n$, surjections, and the finite Coxeter groups $A_n$. A046802 is a binomial transform of the Eulerlian polynomials $E_n(x)$ (cf. A123125 and A008292) with numerous connections to important aspects of algebraic combinatorics and analysis and which contain the h-vectors of the permutohedra / permutahedra--classic polytopes that encode the combinatorics of mulitiplicative inversion of e.g.f.s, umbral inversion of Appell sequences, and surjective mappings. A046802 also enumerates the positroids of the Grassmannians and contains the h-vectors of the stellahedra.
The row polynomials of A046802 are given by $A_n(x;1)$ where $A_n(x;y) = (y + E.(x))^n$ is an Appell sequence in $y$ and the umbral evaluation $E.(x)^n = E_n(x)$ applies. The associated face polynomials of the stellahedra, A248727, are given by $A(1+x;1) = (1+E.(1+x))^n$. In addition, $Sw_n(x) = A_n(-1;1+x) = (1+x+E.(-1))^n$ gives the Swiss-knife polynomials, A119879, related to the Bernoulli, Springer, Euler and other number sequences found at the crossroads of combinatorics, analysis, geometry, and physics. So, there are rich associations here that if connected to tilting modules might provide some mutual enlightenment.
Can anyone prove the author's sugestion is indeed true or provide supporting evidence?
