725 research outputs found
Reciprocals of exponential polynomials and permutation enumeration
We show that the reciprocal of a partial sum with 2m terms of the alternating
exponential series is the exponential generating function for permutations in
which every increasing run has length congruent to 0 or 1 modulo 2m. More
generally we study polynomials whose reciprocals are exponential generating
functions for permutations whose run lengths are restricted to certain
congruence classes, and extend these results to noncommutative symmetric
functions that count words with the same restrictions on run lengths
Applications of the classical umbral calculus
We describe applications of the classical umbral calculus to bilinear
generating functions for polynomial sequences, identities for Bernoulli and
related numbers, and Kummer congruences.Comment: 34 page
A note on Stirling permutations
In this note we generalize an identity of John Riordan and Robert Donaghey
relating the enumerator for Stirling permutations to the Eulerian polynomials.Comment: Originally written in 1978 but not published until no
A simple proof of Andrews's 5F4 evaluation
We give a simple proof of George Andrews's balanced 5F4 evaluation using two
fundamental principles: the nth difference of a polynomial of degree less than
n is zero, and a polynomial of degree n that vanishes at n+1 points is
identically zero
On the Schur function expansion of a symmetric quasi-symmetric function
Egge, Loehr, and Warrington proved a formula for the Schur function expansion
of a symmetric function in terms of its expansion in fundamental
quasi-symmetric functions. Their formula involves the coefficients of a
modified inverse Kostka matrix. Recently Garsia and Remmel gave a simpler
reformulation of Egge, Loehr, and Warrington's result, with a new proof. We
give here a simple proof of Garsia and Remmel's version, using a sign-reversing
involution.Comment: 4 page
A short proof of the Deutsch-Sagan congruence for connected non crossing graphs
We give a short proof, using Lagrange inversion, of a congruence modulo 3 for
the number of connected noncrossing graphs on n vertices that was conjectured
by Emeric Deutsch and Bruce Sagan. A more complicated proof had been given
earlier by S.-P. Eu, S.-C. Liu, and Y.-N. Yeh
Shuffle-compatible permutation statistics
Since the early work of Richard Stanley, it has been observed that several
permutation statistics have a remarkable property with respect to shuffles of
permutations. We formalize this notion of a shuffle-compatible permutation
statistic and introduce the shuffle algebra of a shuffle-compatible permutation
statistic, which encodes the distribution of the statistic over shuffles of
permutations. This paper develops a theory of shuffle-compatibility for descent
statistics (statistics that depend only on the descent set and length) which
has close connections to the theory of -partitions, quasisymmetric
functions, and noncommutative symmetric functions. We use our framework to
prove that many descent statistics are shuffle-compatible and to give explicit
descriptions of their shuffle algebras, thus unifying past results of Stanley,
Gessel, Stembridge, Aguiar-Bergeron-Nyman, and Petersen.Comment: 52 pages. To appear in Adv. Mat
A Combinatorial Interpretation of The Numbers
It is well known that the numbers are integers,
but in general there is no known combinatorial interpretation for them. When
these numbers are the middle binomial coefficients , and
when they are twice the Catalan numbers. In this paper, we give
combinatorial interpretations for these numbers when or 3.Comment: 11 pages, 2 figure
A refinement of Cayley's formula for trees
A proper vertex of a rooted tree with totally ordered vertices is a vertex
that is less than all its proper descendants. We count several kinds of labeled
rooted trees and forests by the number of proper vertices. Our results are all
expressed in terms of the polynomials P_n(a,b,c)=
c(a+(n-1)b+c)(2a+(n-2)b+c)...((n-1)a+b+c) which reduce to (n+1)^{n-1} for
a=b=c=1.
Our study of proper vertices was motivated by A. Postnikov's hook length
formula for binary trees (arXiv:math.CO/0507163), which was also proved by W.
Y. C. Chen and L. L. M. Yang (arXiv:math.CO/0507163) and generalized by R. R.
X. Du and F. Liu (arXiv:math.CO/0501147). Our approach gives a new proof of Du
and Liu's results and gives new hook length formulas.
We also find an interpretation of the polynomials P_n(a,b,c) in terms of
parking functions: we count parking functions according to the number of cars
that park in their preferred parking spaces.Comment: 20 page
A Short Proof of the Zeilberger-Bressoud -Dyson Theorem
We give a formal Laurent series proof of Andrews's -Dyson Conjecture,
first proved by Zeilberger and Bressoud.Comment: 10 pages, minor revisio
- …