2023/06/27 by Vincent Brugidou, Brugidou, Vincent · 1 citation
Computer Science · Mathematics · #05 #13 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Combinatorics (math.CO) #Commutative Algebra (math.AC) #F.2.2 #FOS: Mathematics #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2306.15300
openalex publication_date 2023/06/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let mλ be the monomial symmetric functions, λ being an integer partition of n∈ ℕ∗ . For the specialization corresponding to the q-deformation of the exponential, we prove that each mλ is associated with a polynomial Jλ( q) whose coefficients belong to ℤ. Jλ is a generalization of the case λ=( n) for which J( n) =Jn is the enumerator of tree inversions. Some relations between Jλ and Jn( r) are obtained, these Jn( r) having been defined algebraically in a previous work of the author for n≥ r≥ 1 and being classically combinatorial enumerators with Jn( 1) =Jn. From the calculation by induction of Jλ for n≤ 6, we conjecture that the coefficients of each Jλ are strictly positive and log-concave. As a consequence of Huh's Theorem on the h-vector of matroid complex it is shown that the coefficients of Jn( r) are strictly positive and log-concave, which gives a second argument in favor of these conjectures. It is also proven that the last n-1 coefficients of Jλ are proportional to the first coefficients of column n-r-1 of Pascal's triangle, r being the length of λ. This is a third argument to state the conjectures. The calculation of J( 3,2,1) shows the existence of an obstacle, if one wants to prove the conjectures by application of Huh's theorem cited above.