2,151 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
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
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
The Generating Function of Ternary Trees and Continued Fractions
Michael Somos conjectured a relation between Hankel determinants whose
entries count ternary trees and the number of
certain plane partitions and alternating sign matrices. Tamm evaluated these
determinants by showing that the generating function for these entries has a
continued fraction that is a special case of Gauss's continued fraction for a
quotient of hypergeometric series. We give a systematic application of the
continued fraction method to a number of similar Hankel determinants. We also
describe a simple method for transforming determinants using the generating
function for their entries. In this way we transform Somos's Hankel
determinants to known determinants, and we obtain, up to a power of 3, a Hankel
determinant for the number of alternating sign matrices. We obtain a
combinatorial proof, in terms of nonintersecting paths, of determinant
identities involving the number of ternary trees and more general determinant
identities involving the number of -ary trees.Comment: 44 pages, 12 figure
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
Enumeration of Point-Determining Graphs
Point-determining graphs are graphs in which no two vertices have the same
neighborhoods, co-point-determining graphs are those whose complements are
point-determining, and bi-point-determining graphs are those both
point-determining and co-point-determining. Bicolored point-determining graphs
are point-determining graphs whose vertices are properly colored with white and
black. We use the combinatorial theory of species to enumerate these graphs as
well as the connected cases.Comment: 26 pages, 11 figure
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
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