Three families of q-Lommel polynomials (with Jang Soo Kim, pdf)

**Abstract**Three q-versions of Lommel polynomials are studied. Included are explicit
representations, recurrences, continued fractions, and connections to associated Askey-Wilson
polynomials. Some combinatorial results are given, including a general result when RI
moments coincide with orthogonal polynomial moments. The combinatorial results
use weighted Motzkin paths, Schroder paths, and parallelogram polyominoes.

Orthogonal polynomials of type R_I (with Jang Soo Kim, pdf)

**Abstract**
A combinatorial theory for type R_I orthogonal polynomials is given. The ingredients
include weighted generalized Motzkin paths, moments, continued fractions,
determinants, and histories. Several explicit examples in the Askey scheme are
given.

Dyson's crank and the mex (with B. Hopkins and J. Sellers, pdf)

**Abstract**
Andrews and Newman have recently introduced the notion of the mex of a partition, the smallest positive integer
that is not a part. The concept has been used since at least 2011, though, with connections to Frobenius
symbols. Recently the parity of the mex has been associated to the crank statistic named by Dyson in 1944. In
this note, we extend and strengthen the connection between the crank and mex (along with a new generaliztion of
the mex) by proving a number of properties that naturally relate these partition statistics.

Ratios of Jackson's q-Bessel functions and q-Lommel polynomials (with Jang Soo Kim, pdf)

**Abstract**
In 1993 Delest and Fedou showed that a generating function for connected skew shapes is given as a ratio
J_v+1/J_v
of Jackson's third q-Bessel functions when a parameter v is zero. They conjectured that when v is a
nonnegative integer the coefficients of the generating function are rational functions whose numerator and
denominator are polynomials in q with nonnegative integer coefficients, which is a q-analog of Kishore's 1963
result on Bessel functions. The first main result of this paper is a proof of the conjecture of Delest and
Fedou. The second main result is a refinement of the result of Delest and Fedou: a generating function for
connected skew shapes with bounded diagonals is given as a ratio of q-Lommel polynomials introduced by Koelink
and Swarttouw. It is also shown that the ratio J_{v+1}/J_v has two different continued fraction expressions, which
give respectively a generating function for moments of orthogonal polynomials of type R_I and a generating
function for moments of usual orthogonal polynomials. Orthogonal polynomial techniques due to Flajolet and
Viennot are used.

Some problems An informal summary of problems I have failed to do, but I have not given up. June 2019.

Marking and shifting a part in partition theorems (with K. O'Hara)

** Abstract**
Refined versions, analytic and combinatorial, are given for
classical integer partition theorems. The examples include the Rogers-Ramanujan
identities, the G"ollnitz-Gordon identities, Euler's odd=distinct theorem, and
the Andrews-Gordon identities. Generalizations of each of these theorems are given where a
single part is ``marked" or weighted. This allows a single
part to be replaced by a new larger part, ``shifting" a part,
and analogous combinatorial results are given in each case.
Versions are also given for marking a sum of parts.

Character sheaves for symmetric pairs I wrote an appendix to this paper by K. Vilonen and T. Xue

Orthogonal polynomials and Smith normal form (with Alex Miller, pdf)

** Abstract**
Smith normal form evaluations found by Bessenrodt and Stanley for some Hankel matrices of q-Catalan
numbers are proven in two ways. One argument generalizes the Bessenrodt-Stanley results for the
Smith normal form of a certain multivariate matrix that refines one studied by Berlekamp, Carlitz,
Roselle, and Scoville. The second argument, which uses orthogonal polynomials, generalizes to a
number of other Hankel matrices, Toeplitz matrices, and Gram matrices. It gives new results for
q-Catalan numbers, q-Motzkin numbers, q-Schroder numbers, q-Stirling numbers, q-matching numbers,
q-factorials, q-double factorials, as well as generating functions for permutations with eight
statistics.

On q-integrals over order polytopes (with Jang Soo Kim, pdf)

** Abstract**
A combinatorial study of multiple q-integrals is developed. This includes a q-volume of a convex
polytope, which depends upon the order of q-integration. A multiple q-integral over an order polytope of
a poset is interpreted as a generating function of linear extensions of the poset. Specific
modifications of posets are shown to give predictable changes in q-integrals over their respective order
polytopes. This method is used to combinatorially evaluate some generalized q-beta integrals. One such
application is a combinatorial interpretation of a q-Selberg integral. New generating functions for
generalized Gelfand-Tsetlin patterns and reverse plane partitions are established. A q-analogue to a
well known result in Ehrhart theory is generalized using q-volumes and q-Ehrhart polynomials.

Binomial Andrews-Gordon-Bressoud identities (pdf)

** Abstract**
Binomial versions of the Andrews-Gordon-Bressoud identities are given.

On the cohomology of Fano varieties and the Springer correspondence (pdf) I wrote an appendix to this paper by T.-H. Chen, K. Vilonen, and T. Xue

Asymptotics of the number of involutions in finite classical groups (with Jason Fulman and Robert Guralnick, pdf)

** Abstract**
Answering a question of Geoff Robinson, we compute the large n
limiting proportion of i(n,q)/q^[n^2/2], where i(n,q) denotes the number of
involutions in GL(n,q). We give similar results for the finite unitary,
symplectic, and orthogonal groups, in both odd and even characteristic.
At the heart of this work are certain new ``sum=product" identities. Our self-contained
treatment of the enumeration of involutions in even characteristic symplectic and
orthogonal groups may also be of interest.

Moments of orthogonal polynomials and combinatorics (with Sylvie Corteel and Jang Soo Kim, pdf)

**Abstract:**
This paper is a survey on combinatorics of moments of orthogonal polynomials and linearization
coefficients. This area was started by the seminal work of Flajolet followed by Viennot at the beginning of the 1980's. Over
the last 30 years, several tools were conceived to extract the combinatorics and compute these moments. A
survey of these techniques is presented, with applications to polynomials in the Askey scheme.

On the distribution of the number of fixed vectors for the finite classical groups (with Jason Fulman, pdf)

** Abstract**
Motivated by analogous results for the symmetric group and compact Lie groups, we study the
distribution of the number of fixed vectors of a random element of a finite classical group. We
determine the limiting moments of these distributions, and find exactly how large the rank of the
group has to be in order for the moment to stabilize to its limiting value. The proofs require a
subtle use of some q-series identities. We also point out connections with orthogonal
polynomials.

The combinatorics of associated Laguerre polynomials (with Jang Soo Kim, pdf)

** Abstract**
The explicit double sum for the associated Laguerre polynomials is
derived combinatorially. The moments are described using certain
statistics on permutations and permutation tableaux. Another
derivation of the double sum is provided using only the moment
generating function.

Refinements of the Rogers-Ramanujan identities The Official Journal link (with K. O'Hara, pdf)

** Abstract**
Refinements of the classical Rogers-Ramanujan
identities are given in which some parts are weighted. Combinatorial interpretations
refining MacMahon's results are corollaries.

Orthogonality of very well-poised series (with M. Ismail and E. Rains, pdf)

** Abstract**
Rodrigues formulas for very well-poised basic hypergeometric series of any order are
given.
Orthogonality relations are found for rational functions which generalize Rahman's
10\phi9
biorthogonal rational functions.

Invariants of GL_{n}(F_{
q}) in polynomials mod
Frobenius powers
(with J. Lewis, V. Reiner, pdf)

** Abstract**
Conjectures are given for Hilbert series related to polynomial
invariants of finite general linear groups, one for invariants mod
Frobenius powers of the irrelevant ideal, one for cofixed spaces of
polynomials.

Expansions in Askey-Wilson polynomials (with M. Ismail, pdf)

** Abstract**
We give a general expansion formula of functions in the Askey--Wilson polynomials and
using the
Askey--Wilson orthogonality we evaluate several integrals. Moreover we give a
general expansion formula of functions in polynomials of Askey--Wilson type, which are
not necessarily orthogonal.
Limiting cases give expansions in little and big q-Jacobi type polynomials. We also
give a
new generating function for Askey--Wilson polynomials and a new evaluation for
specialized
Askey--Wilson polynomials.

What is cyclic sieving? (with V. Reiner and D. White, pdf)

(This is a self-explanatory blurb for the Notices of the AMS.)

The q=-1 phenomenon via homology concentration (with P. Hersh and J. Shareshian, pdf)

** Abstract**
We introduce a homological approach to exhibiting instances of
Stembridge's q=-1
phenomenon. This approach is shown to explain two important instances
of the
phenomenon, namely that of partitions whose Ferrers diagrams fit in a
rectangle
of fixed size and that of plane partitions fitting in a box of fixed
size.
A more general framework of invariant and coinvariant complexes with
coefficients taken mod 2 is developed, and as a part of this story
an analogous homological result for necklaces is conjectured.

Bootstrapping and Askey-Wilson polynomials (with Jang Soo Kim, pdf)

** Abstract**
The mixed moments for the Askey-Wilson polynomials
are found using a bootstrapping method and connection coefficients. A
similar
bootstrapping idea on generating functions gives a new Askey-Wilson
generating function. Modified generating
functions of orthogonal polynomials are shown to generate
polynomials satisfying recurrences of known degree greater than
three.
An important special case of this hierarchy is a polynomial which
satisfies a four
term recurrence, and its combinatorics is studied.

Reflection factorizations for Singer cycles (with Joel Lewis and Vic Reiner, pdf)

** Abstract**
The number of shortest factorizations into reflections for a
Singer cycle in GL_n(F_q) is shown to be (q^n-1)^{n-1}.
Formulas counting factorizations of any length, and counting those with reflections of fixed conjugacy classes are also given.

Orthogonal basic hypergeometric Laurent polynomials (with Mourad Ismail, pdf)

** Abstract**
The Askey-Wilson polynomials are orthogonal polynomials in
x = \cos \theta, which
are given as a terminating _4\phi_3 basic hypergeometric series.
The non-symmetric Askey-Wilson polynomials are Laurent polynomials in
z=e^{i\theta}, which are given as a sum of two terminating
4\phi_3's.
They satisfy a biorthogonality relation. In this paper new orthogonality
relations for single 4\phi_3's which are Laurent polynomials in z
are given,
which imply the non-symmetric Askey-Wilson biorthogonality. These
results include
discrete orthogonality relations. They can be considered as a classical
analytic
study of the results for non-symmetric
Askey-Wilson polynomials which were previously obtained by affine Hecke
algebra techniques.

Applications of q-Taylor theorems (with Mourad Ismail) (ps) dvi version

**Abstract** We establish two new q-analogues of a Taylor
series expansion for polynomials using special Askey-Wilson polynomial bases.
Combining these expansions with an earlier expansion theorem we
derive inverse relations and evaluate certain linearization
coefficients. Byproducts
include new summation theorems, new results on a
q-exponential function, and quadratic transformations for q-series.

Moments of Askey-Wilson polynomials (with Jang Soo Kim, pdf )

** Abstract**
New formulas for the n^th moment \mu_n(a,b,c,d;q) of the
Askey-Wilson
polynomials are given. These are derived using analytic techniques, and
by considering
three combinatorial models for the moments: Motzkin paths, matchings,
and
staircase tableaux. A related positivity theorem is given and another
one is conjectured.

The negative q-binomial (with S. Fu, V. Reiner, and N. Thiem, pdf)

** Abstract** Interpretations for the
q-binomial coefficient evaluated at -q are
discussed. A (q,t)-version is established,
including an instance of a cyclic sieving phenomenon
involving unitary spaces.

Some combinatorial and analytical identities (with M. Ismail)

** Abstract** We give new proofs and explain the origin of several
combinatorial identities of Fu and Lascoux, Dilcher, Prodinger, and
Uchimura. We use the theory of basic hypergeometric functions, and
generalize these identities. We also exploit the theory of polynomial
expansions in the Wilson
and Askey-Wilson bases to derive new identities which are not in the
hierarchy of basic hypergeometric series.
We demonstrate that a Lagrange interpolation formula always leads to
very-well-poised
basic hypergeometric series. As applications we prove that the Watson
transformation
of a balanced 4\phi_3 to a very-well-poised 8\phi_7 is
equivalent to the
Rodrigues-type formula for the Askey-Wilson polynomials.

Formulae for Askey-Wilson moments and enumeration of staircase tableaux (with S. Corteel, R. Stanley, and L. Williams, pdf)

** Abstract** We explain how the moments of the (weight function
of the) Askey Wilson polynomials
are related to the enumeration
of the staircase tableaux
introduced by the first and fourth authors.
This gives us a direct combinatorial formula
for these moments, which is related to, but more
elegant than the formula previously.
Then we use techniques developed by Ismail and
the third author to give explicit formulae
for these moments and for the enumeration of staircase tableaux.
Finally we study the enumeration of staircase tableaux at various
specializations of the parameterizations; for example, we obtain
the Catalan numbers, Fibonacci numbers, Eulerian numbers,
the number of permutations, and the number of matchings.

The combinatorics of Al-Salam-Chihara q-Laguerre polynomials (with A. Kasraoui and J. Zeng, pdf)

** Abstract** We decribe various aspects of the Al-Salam-Chihara
q-Laguerre polynomials.
These include combinatorial descriptions of the polynomials, the
moments, the
orthogonality relation and a combinatorial interpretation of the
linearization coefficients.

(q,t)-analogues and
GL_{n}(F_{q})
(with V. Reiner, pdf)

** Abstract** We start with a (q,t)-generalization of a binomial
coefficient. It can be viewed as a polynomial in t that depends upon an integer
q, with combinatorial interpretations when q is a positive integer, and
algebraic interpretations when q is the order of a finite field.
These (q,t)-binomial coefficients and their interpretations generalize further in two directions, one
relating to column-strict tableaux and Macdonald's ``7^{th} variation'' of
Schur functions, the other relating to permutation statistics and
Hilbert series from the invariant theory of GL_{n}(F_{q}).

The combinatorics of the Al-Salam-Chihara q-Charlier polynomials (with D. Kim and J. Zeng) (pdf)

** Abstract** We describe various aspects of the
Al-Salam-Chihara q-Charlier polynomials. These include combinatorial descriptions of the polynomials, the moments, the orthogonality
relation and a combinatorial proof of Anshelevich's recent result
on the linearization coefficients.

Note on 1-crossing partitions (with M. Bergerson, A. Miller, A. Pliml, V. Reiner, P. Shearer, and N. Switala) (pdf)

Bimahonian distributions (with H. Barcelo and V. Reiner) (pdf)

q-analogues of Euler's Odd=Distinct Theorem (pdf)

** Abstract** Two q-analogues of Euler's theorem on
integer partitions with odd or distinct parts are given. A q-lecture hall theorem is given.

Block inclusions and cores of partitions (with B. Olsson) (pdf)

** Abstract**
Necessary and sufficient conditions are given for an s-block of integer
partitions to be contained in a t-block. The generating function for
such partitions is found analytically, and also bijectively, using the
notion of an (s, t)-abacus. The largest partition which is both an
s-core and a t-core is explicitly given.

Springer's theorem for modular coinvariants of GL_n(F_q) (with V. Reiner and P. Webb) (ps)

** Abstract** Two related results are proven in the modular invariant
theory of GL_n(F_q). The first is a finite field analogue of a result of
Springer on coinvariants of the symmetric group in characteristic zero.
The second result is a related statement about parabolic invariants and
coinvariants.

Ramanujan Continued Fractions Via Orthogonal Polynomials (with M. Ismail) (pdf)

** Abstract**
Some Ramanujan continued fractions are evaluated using
asymptotics of polynomials orthogonal with respect to
measures with absolutely continuous components.

Springer's regular elements over arbitrary fields (with V. Reiner and P. Webb) (pdf)

** Abstract**
Springer's theory of regular elements in complex reflection groups
is generalized to arbitrary fields. Consequences for the
* cyclic sieving phenomenon* in combinatorics are discussed.

The cyclic sieving phenomenon (with V. Reiner and D. White) (pdf)

** Abstract** The cyclic sieving phenomenon is defined for generating
functions of a set affording a cyclic group action, generalizing
Stembridge's q=-1 phenomenon. The phenomenon is shown to appear in
various situations, involving q-binomial coefficients, Polya theory,
polygon dissections, non-crossing partitions, finite reflection groups,
and some finite field q-analogues.

Summable sums of hypergeometric series (ps)

** Abstract**
New expansions for certain 2F1's
as a sum of r higher order hypergeometric series are given. When
specialized
to the binomial theorem, these r hypergeometric series sum. The
results represent cubic and higher order transformations, and only
Vandermonde's
theorem is necessary for the elementary proof. Some q-analogues are
also given.

q-Taylor theorems, polynomial expansions, and interpolation of entire functions (with Mourad Ismail, dvi) ps version

**Abstract**
We establish q-analogues of Taylor series expansions in special
polynomial bases for functions where ln M(r;f) grows like
ln^2 r. This solves the problem of constructing such entire functions
from their values at [aq^n+ q^{-n}/a]/2, for 0 < q < 1. Our technique
is constructive and gives an explicit representation of the sought
entire function. Applications to q-series identities are given.

The Charney-Davis quantity for certain graded posets (with V. Reiner and V. Welker, ps) dvi latex pdf

**Abstract**
Given a naturally labelled graded poset P with r ranks,
the alternating sum

W(P,-1):=\sum_{w \in \JH(P)} (-1)^{\des(w)}

is an instance of a quantity occurring in the Charney-Davis Conjecture on flag simplicial spheres. When |P|-r is odd it vanishes. When |P|-r is even and P satisfies the Neggers-Stanley Conjecture, it has sign (-1)^{\frac{|P|-r}{2}}.

We interpret this quantity combinatorially for several classes of graded posets P, including certain disjoint unions of chains and products of chains. These interpretations involve alternating multiset permutations, Baxter permutations, Catalan numbers, and Franel numbers.

Enumeration and special functions (ps) pdf version(pdf is the latest version) dvi version

**Abstract** These are notes for the 5 talks I gave at
the August 12-16, 2002 Euro Summer School in OPSF at Leuven.

Proof of a monotonicity conjecture (with Thomas Prellberg, ps)

**Abstract** A monotonicity conjecture of Friedman, Joichi and
Stanton is established.

Tribasic Integrals and identities of Rogers-Ramanujan type (with Mourad Ismail) (dvi)

**Abstract** Some general integrals involving three
bases are evaluated as infinite products using complex analysis.
Many special cases of these integrals
may be evaluated in another way to find infinite sum
representations for these
infinite products. The resulting identities are identities of Rogers-Ramanujan type.
Some integer partition interpretations of these identities are given.
Generalizations of the Rogers-Ramanujan type identities
involving polynomials are given,
again as corollaries of integral evaluations.

Fake Gaussian sequences (ps) pdf version

**Abstract** This is not a paper, but a report.
Some positivity conjectures are made generalizing known
results for Gaussian posets. A summary of partial results is given.