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| Eulerian polynomial | |
|---|---|
| Name | Eulerian polynomial |
| Field | Mathematics |
| Introduced | 18th century |
| Notable | Leonhard Euler |
Eulerian polynomial is a family of polynomials that encode permutation statistics and appear across algebraic combinatorics, enumerative combinatorics, and analytic number theory. These polynomials, classically attributed to Leonhard Euler, relate to permutation descents, generate connections with Carl Gustav Jacobi-type identities, and surface in the study of Bernoulli numbers, Riemann zeta function, and special functions studied by Joseph Fourier and Adrien-Marie Legendre. They have influenced work by Richard Stanley, Philippe Flajolet, George Pólya, and researchers at institutions such as Princeton University and University of Cambridge.
For a positive integer n the Eulerian polynomial A_n(t) is defined by the descent statistic on permutations of {1,...,n}, summing t^{des(π)} over all permutations π in the symmetric group S_n; the coefficients A(n,k) count permutations with k descents. The classical formulation was studied by Leonhard Euler and later systematized in texts by Cauchy and Augustin-Louis Cauchy-era combinatorialists. Variants appeared in correspondence involving Joseph-Louis Lagrange and in lectures at École Polytechnique by contemporaries.
Eulerian polynomials satisfy symmetry and unimodality properties proved by methods used by Paul Erdős, Gábor Szegő, and George Pólya. The coefficients A(n,k) obey recurrence relations discovered in Leonhard Euler's notebooks and later formalized by J. Riordan and Doron Zeilberger. The polynomials have integer coefficients, degree n−1, and leading coefficient 1 for n≥1. Classical identities connect Eulerian numbers with Stirling numbers of the first kind studied by James Stirling and with Bernoulli numbers investigated by Jakob Bernoulli and Jacob Bernoulli.
Exponential and ordinary generating functions for Eulerian polynomials were developed in correspondence among Leonhard Euler, Adrien-Marie Legendre, and later exposited by Richard Stanley and Philippe Flajolet. The exponential generating function ties to the exponential series central to Leonhard Euler's analytic work and to formulas appearing in treatises by Leonard Euler's contemporaries. Recurrences include a three-term recurrence used by Dorothy Mahalanobis-era statisticians and linear relations employed in work at Harvard University and Massachusetts Institute of Technology combinatorics seminars. Connections exist with identities attributed to Carl Friedrich Gauss and transformations studied by Niels Henrik Abel.
Classical combinatorial interpretations relate Eulerian polynomials to descents in permutations of S_n, excedances in studies by André-type enumerators, and runs studied in enumerative problems by Percy Alexander MacMahon. Bijective proofs involve involutions and Foata's transformations linked to research by Dominique Foata and Gian-Carlo Rota. Interpretations extend to objects in Young tableau theory treated by Alfred Young and to poset Eulerian properties explored by Richard Stanley and Gian-Carlo Rota at conferences at University of Florence and University of Rome.
The location and distribution of roots of Eulerian polynomials connect to questions in complex analysis explored by Bernhard Riemann-inspired research, and to stability criteria used by Oskar Perron and Issai Schur. Real-rootedness results in special cases were proved using techniques from work by Richard Stanley and Brenti, and interlacing properties relate to linear operators studied by Stefan Banach and John von Neumann. Asymptotic behavior of coefficients links to central limit theorems considered by Andrey Kolmogorov and Paul Lévy and to saddle-point methods employed by H. Jeffreys and analysts at University of Göttingen.
Eulerian polynomials generalize to q-analogues developed in research by Gasper Gosper-adjacent communities, to multivariate versions studied by Gian-Carlo Rota and Richard Stanley, and to Eulerian distributions on Coxeter groups investigated by Humphreys and others in Lie-theoretic contexts at Princeton University and University of Chicago. They appear in Hilbert series computations in algebraic geometry seminars at Institut des Hautes Études Scientifiques and in representation-theoretic studies tied to Frobenius characters. Modern applications touch on algorithmic analysis in work by Donald Knuth and on stochastic models considered by William Feller and researchers at Bell Labs.
Category:Polynomials