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## Phylogenetic Invariants for $${\mathbb{Z}_3}$$ Z 3 Scheme-Theoretically

### Annals of Combinatorics (2016-09-01) 20: 549-568 , September 01, 2016

We study phylogenetic invariants of general group-based models of evolution with group of symmetries
$${\mathbb{Z}_3}$$
. We prove that complex projective schemes corresponding to the ideal *I* of phylogenetic invariants of such a model and to its subideal
$${I'}$$
generated by elements of degree at most 3 are the same. This is motivated by a conjecture of Sturmfels and Sullivant [14, Conj. 29], which would imply that
$${I = I'}$$
.

## On the Probability that Certain Compositions Have the Same Number Of Parts

### Annals of Combinatorics (2010-09-01) 14: 291-306 , September 01, 2010

We compute the asymptotic probability that two randomly selected compositions of *n* into parts equal to *a* or *b* have the same number of parts. In addition, we provide bijections in the case of parts of sizes 1 and 2 with weighted lattice paths and central Whitney numbers of fence posets. Explicit algebraic generating functions and asymptotic probabilities are also computed in the case of pairs of compositions of *n* into parts at least *d*, for any fixed natural number *d*.

## A Criterion for Bases of the Ring of Symmetric Functions

### Annals of Combinatorics (2005-12-01) 9: 495-499 , December 01, 2005

### Abstract.

We give a criterion for bases of the ring of symmetric functions in *n* indeterminates over a commutative ring *R* with identity. A related algorithm is presented in the last section.

## A New Family of Positive Integers

### Annals of Combinatorics (2002-12-01) 6: 399-405 , December 01, 2002

### Abstract.

Let *n, p, k* be three integers with
$ 1 \leq K \leq n \quad\mathrm{and}\quad 0\leq p\leq n $
. We prove that the numbers
$ {n \choose k} _3 F_2 (1-k, -p, p-n; 1,1-n; 1) $
are positive integers which generalize the classical binomial coefficients
$ n \choose p $
. We give two generating functions for these integers, and a straightforward application.

## The Mix of a Regular Polytype with a Face

### Annals of Combinatorics (2002-08-01) 6: 77-86 , August 01, 2002

### Abstract.

Mixing is an operation which yields subgroups generated by involutions of a larger group of the same kind. When it is applied to the product of the automorphism groups of two regular polytopes, one talks about the mix of the polytopes. This paper is concerned with conditions under which the mix of two regular polytopes is again a regular polytope. In the important special case of a mix of a polytope with one of its faces, fairly general results about the polytopality of the mix are obtained. In particular, the case when the mix is isomorphic to the original polytope is characterized.

## Permutation Diagrams, Fixed Points and Kazhdan-Lusztig R-Polynomials

### Annals of Combinatorics (2006-12-01) 10: 369-387 , December 01, 2006

### Abstract.

In this paper, we give an algorithm for computing the Kazhdan-Lusztig *R*-polynomials in the symmetric group. The algorithm is described in terms of permutation diagrams. In particular we focus on how the computation of the polynomial is affected by certain fixed points. As a consequence of our methods, we obtain explicit formulas for the *R*-polynomials associated with some general classes of intervals, generalizing results of Brenti and Pagliacci.

## Hook Lengths and 3-Cores

### Annals of Combinatorics (2011-06-01) 15: 305-312 , June 01, 2011

Recently, the first author generalized a formula of Nekrasov and Okounkov which gives a combinatorial formula, in terms of hook lengths of partitions, for the coefficients of certain power series. In the course of this investigation, he conjectured that *a*(*n*) = 0 if and only if *b*(*n*) = 0, where integers *a*(*n*) and *b*(*n*) are defined by
$$\sum^{\infty}_{n=0}\, a(n)x^{n} := \prod^{\infty}_{n=1} \, (1-x^{n})^8,$$
$$\sum^{\infty}_{n=0} \, b(n)x^{n} := \prod^{\infty}_{n=1} \, \frac{(1-x^{3n})^{3}}{1-x^n} .$$
The numbers *a*(*n*) are given in terms of hook lengths of partitions, while *b*(*n*) equals the number of 3-core partitions of *n*. Here we prove this conjecture.

## On Principal Hook Length Partitions and Durfee Sizes in Skew Characters

### Annals of Combinatorics (2011-03-01) 15: 81-94 , March 01, 2011

We construct for a given arbitrary skew diagram $${\mathcal A}$$ all partitions ν with maximal principal hook lengths among all partitions with [ν] appearing in [ $${\mathcal A}$$ ]. Furthermore, we show that these are also partitions with minimal Durfee size. We use this to give the maximal Durfee size for [ν] appearing in [ $${\mathcal A}$$ ] for the cases when $${\mathcal A}$$ decays into two partitions and for some special cases of $${\mathcal A}$$ . We also deduce necessary conditions for two skew diagrams to represent the same skew character.

## The Erdős-Ko-Rado Property for Some 2-Transitive Groups

### Annals of Combinatorics (2015-12-01) 19: 621-640 , December 01, 2015

A subset *S* of a group *G* ≤ Sym(*n*) is intersecting if for any pair of permutations
$${\pi, \sigma \in S}$$
there is an
$${i \in {1, 2, . . . , n}}$$
such that
$${\pi (i) = \sigma (i)}$$
. It has been shown, using an algebraic approach, that the largest intersecting sets in each of Sym(*n*), Alt(*n*), and PGL(2, *q*) are exactly the cosets of the point-stabilizers. In this paper, we show how this approach can be applied more generally to many 2-transitive groups. We then apply this method to the Mathieu groups and to all 2-transitive groups with degree no more than 20.

## An Improvement of the Godsil Bound

### Annals of Combinatorics (2002-08-01) 6: 33-44 , August 01, 2002

### Abstract.

In 1988, Godsil [12] showed that for a distance-regular graph with valency *k* and an eigenvalue with multiplicity
$ m\geq 3 $
the diameter *d* is bounded by
$ 3m - 4 $
. In this note, we show that
$ d $<$ m + 2+\log_5 (m) $
. Furthermore, we show that if the numerical girth is at least 6, then
$ d \leq m + 6 $
. Finally, we show that if the numerical girth is at least 12 then
$ kd $<$ 48m $
.