Research Statement
Samantha Dahlberg
My field of study is enumerative and algebraic combinatorics. One way to define combinatorics is
as the study of discrete structures. In contrast to other fields of mathematics combinatorics is highly
accessible at its basic level, but displays a depth which is more like a spider-web of connections to
many other fields of mathematics and interesting relevant results. Combinatorics can be found in
many other fields including algebraic geometry, computer science, topology, representation theory
and algebra.
My work in enumerative combinatorics has primarily been focused on pattern avoidance in
various combinatorial objects. I have studied pattern avoidance in involutions, a subset of the
symmetric group. In another project with a Research Experience for Undergraduates (REU) group
during summer 2014 I studied pattern avoidance in set partitions and restricted growth functions.
The goal of all these projects was to investigate the generating functions constructed from the
combination of a statistic on the combinatorial objects in question and a specified avoidance class.
This type of research has produced a massive number of results in recent years.
A more recent portion of my work has been on Hopf algebras. The study of these from a
combinatorial perspective is a recent development. Many Hopf algebras have bases which are in
bijection with combinatorial objects like set partitions. The operations in these algebras are usually
basic operations on these objects and as result we find that the algebra itself encodes information
about the combinatorial objects. This can lead to interesting results with enlightening proofs. My
work has been focused on the Hopf algebra of the symmetric functions in noncommuting variables,
NCSym, and its antipode.
Pattern avoidance in Involutions
Pattern avoidance is a younger field having started in the early 1960s and originates not from
the field of mathematics, but from the field of computer science. Donald Knuth is often cited as
the founder of the area.
Let Sn be the symmetric group on [n] = {1, 2, . . . , n}. Two words are order-isomorphic if
all corresponding pairs of numbers share the same relative order like 6935 does with 3412. We
would say that the permutation 351624 contains the pattern 213 since it has the order-isomorphic
subsequence 316, but it avoids the pattern 321 since there is no length three subsequence which
strictly decreases. Let Sn (π) be the set of length n permutations which avoid π. Knuth found that
the stack-sortable permutations
were the same permutations which avoided the pattern 231, and
2n
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additionally |Sn (π)| = n+1
n the nth Catalan number for all π ∈ S3 .
Our investigation looked at the subset of involutions In and In (π) = In ∩ Sn (π). Simion and
Schmidt in [?] found the cardinalities for the sets of involutions which avoid a length three pattern.
Working from these cardinalities and following the same route as Dokos et. al. [?], who did the same
study but with Sn , we investigated the distribution of the major index and the number of inversions
as a generating function over classes of involutions which avoid one or more patterns of length three.
The major index is defined from the descent set of a permutation, Des(π) = {i : π(i) > π(i + 1)}.
For our example Des(351624) = {2, 4}. The major index or maj is the sum of all the elements in
Des(π). For our example maj(351624) = 6. The generating functions we investigated were defined
by
X
M In (π; q) = M In (π) =
q maj(ι)
ι∈In (π)
or M In (S) when considering the avoidance class of multiple patterns S ⊂ Sk . A motivation behind
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P
studying this statistic and these generating functions is the result σ∈Sn q maj(σ) = [n]q ! where [n]q !
is the standard q-analogue of n! defined by [n]q = 1 + q + · · · + q n−1 and [n]q ! = [n]q [n − 1]q . . . [1]q .
We characterized nearly all the functions In (S) for S ⊂ S3 and noted some nice symmetries and
other properties. One particularly nice property we found is in regards to the standard q-analogue
of the binomial coefficients
[n]q !
n
=
.
k q
[k]q ![n − k]q !
These polynomials were first studied by Euler and have found many interpretations and appearances
since. It turns out that
n
M In (321) =
.
dn/2e q
This result was proven independently by Barnabei et al. [?] whose proof gives a connection to
hook decompositions. Our proof has the advantage of being shorter and gives a connection to the
concept of a core. The core is a concept that originated in the study of posets and symmetricchain decompositions, and our use of it is new and quite different from the original. Our proof
can be easily generalized to give another interpretation for the general q-analogue for the binomial
coefficient, not just the central one.
We also have proven some symmetries between pairs π, σ ∈ S3 of patterns of the form
n
q ( 2 ) M In (π; q −1 ) = M In (σ; q).
Such pairs of patterns include 123 and 321, 213 and 132, and 312 and 231.
Pattern avoidance in Set Partitions and RGFs
Another branch of pattern avoidance comes from considering pattern avoidance in other structures. Other objects considered include set partitions and restricted growth functions. A set
partition of [n] is a tuple of pairwise disjoint subsets Bi such that ∪Bi = [n]. The standard form
of a partition is written B = B1 /B2 / . . . /Bk so that min Bi < min Bi+1 like B = 18/267/35/4.
Consider the subpartition 26/8. Its standardization is 12/3, the partition we get from replacing the
ith smallest letter with i. We say that the partition B contains the pattern 12/3 since it contains
the subpartition 26/8 which standardizes 12/3. The other objects we considered were restricted
growth functions (RGFs) which are words w = w1 w2 , . . . wn of positive integers such that w1 = 1
and
wi ≤ max{w1 , w2 , . . . , wi−1 } + 1.
Let Rn be the collection of length n RGFs. We say that the RGF w = 12343221 contains the
pattern 121 since it has the subword 242 which standardizes to 121, but w avoids 112 since there
is no subword which standardizes to 112. Restricted growth functions are in bijection with set
partitions B = B1 /B2 / . . . /Bk of [n] with the identification i ∈ Bj if and only if wi = j. For
example, the associated set partition of w = 12343221 is B = 18/267/35/4. These two notions of
pattern avoidance are different. While w avoids 112 the partition B has the pattern 12/3.
During summer 2014 I worked with my advisor and a group of undergraduates in an REU
program. Our topic was pattern avoidance in set partitions and restricted growth functions. Our
project was to determine the generating functions using Wachs and White’s four statistics on
RGFs [?].
Wachs and White’s statistics on RGFs are left-bigger, left-smaller, right-bigger, right-smaller
abbreviated lb, ls, rb and rs respectfully. The left-bigger statistic considers each letter wi in the
RGF w and counts the number of distinct numbers left and bigger than it. The sum of all these
calculation is lb. The other statistics are defined similarly.
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The first section of this project was to study pattern avoidance in set partitions. For nearly
all the avoidance classes of partitions of [3] we were able to determine the four-variate generating
function incorporating all of Wachs and White’s statistics. We were also able to describe the fourvariate generating function for pairs and triples of partitions of [3]. This paper has been accepted
for publication in Discrete Mathematics [?].
We also investigated the similar, but different, notion of pattern avoidance in RGFs. We considered all length n RGFs which avoid v ∈ Rk and called this set Rn (v). For nearly all pairs and
triples V of length three patterns we were able to determine the four-variate generating function.
The single-variate version is defined by
X
q lb(w)
LBn (V ) = LBn (V ; q) =
w∈Rn (B)
with LSn (V ), RBn (V ) and RSn (V ) defined similarly. For single patterns we were able to characterize most of the generating functions for the four statistics separately and prove all of the
equalities we saw. One such proof defines a bijection which makes use of Young diagrams and hook
decompositions.
Some avoidance classes gave particularly interesting generating functions. The cardinalities of
Rn (1212) and Rn (1221) are both the nth Catalan number, and these two sets are well studied and
are called the non-crossing and non-nesting set partitions, respectively. Some of the generating
functions like
LBn (1212) = LBn (1221) = RSn (1212)
gave a familiar and previously studied Catalan analogue, C0 (q) = 1 and
Cn (q) = 2Cn−1 (q) +
n−2
X
q k Ck (q)Cn−k−1 (q).
k=1
We proved all these equalities by establishing a bijection from our sets of RGFs to two-colored
Motzkin paths where the area under the Motzkin path was equal to the appropriate statistic on
the RGF.
Hopf Algebras
The study of Hopf algebras came from the field of algebraic topology around the 1950s. Most
of the research on combinatorial Hopf algebras didn’t appear in papers until the 1990s. One
particular Hopf algebra that has generated a lot of interest is NCSym, the algebra of symmetric
functions in non-commuting variables. That is, the set of all formal power series f ∈ Q x in
non-commuting variables x = {x1 , x2 , . . . } such that for all π ∈ Sn ,
f (x1 , x2 , . . . ) = f (xπ(1) , xπ(2) , . . . ).
For example x1 x2 x1 + x2 x1 x2 + x1 x3 x1 + x3 x1 x3 + x2 x3 x2 + x3 x2 x3 + · · · ∈ NCSym. All the different
bases of NCSym are in bijection with set partitions.
My first work on Hopf algebras started after wondering if there was a Hopf algebra with a basis
set indexed by involutions. Inside our student seminar group the answer to this was not known,
so I constructed two Hopf algebras on involutions. For one of the two I found a cancellationfree antipode formula. The proof used Takeuchi’s formula for the antipode and a sign-reversing
involution, a method first used by Benedetti and Sagan [?].
Thanks to a conversation with Aaron Lauve we found that the two Hopf algebras were really
sub-Hopf algebras of NCSym using two different bases. When we restrict the power-sum basis and
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the homogeneous basis of NCSym to set partitions with blocks of size at most two we obtained my
two Hopf algebras on involutions. The identification between the partitions and the involutions is
that the blocks of size two in the set partition are associated to two-cycles in the involution and
blocks of size one are associated to fixed points. The cancellation-free antipode formula which I
had for the involution case generalized to the larger Hopf algebra without any trouble, and we had
then a cancellation-free antipode formula for NCSym. Before this Lauve and Mastnak [?] had a
formula for the antipode of NCSym in the power-sum basis. However, their formula still had terms
in the summation which canceled.
It was recently in October 2015 at an AMS sectional meeting where we learned that a cancellationfree formula was already known. This formula comes from a paper by Baker-Jarvis, Bergeron,
and Thiem [?] in which they use the Hopf monoid structure and super characters to get their
cancellation-free antipode formula but which appears in a different form from the one I derived.
This information has led to some questions that we hope to answer in the future and are described
in the section below.
Future Plans for Pattern Avoidance
1. We have noted that in S3 there are three pairs of patterns π, σ which satisfy the symmetry
n
q ( 2 ) M In (π; q −1 ) = M In (σ; q). By a quick analysis of patterns in S4 we can see that there
are far fewer pairs which satisfy the symmetry. The question is this: What pairs in S4 display
this symmetry, and can we generalize to Sk .
2. Another direction pattern avoidance has taken is called generalized pattern avoidance which
was first defined by Babson and Steingrı́msson [?]. A permutation π has a generalized pattern
like 321 if contains the pattern 321 but we require that the underlined portion appear in
adjacent indices in π. Some of the avoidance classes of generalized patterns share an same
enumeration with involutions, partitions or Motzkin paths.
We have started to consider generalized patterns in involutions. So far we have enumerated
In (321) and In (321) and have also proven some other results about these avoidance classes.
The plan here would be to classify the avoidance classes of involutions for all generalized
patterns of length three.
3. Another plan is to continue exploring various generating functions over Rn (1212) and Rn (1221).
We have come in contact with Jay Pantone who has a computer program which might generate
some conjectures for the recurrences of these functions.
Future Plans for Hopf Algebras
As mentioned in the Hopf algebra section above Baker-Jarvis, Bergeron, and Thiem have a
cancellation-free formula for the antipode in the power sum basis. There are a number of questions
which we would like to answer in regards to this.
1. How can one give a direct proof that the two cancellation-free formulas are equivalent?
2. Can a sign-reversing involution provide a cancellation-free formula in the monoid case?
3. Can sign-reversing involutions be used to find cancellation-free formulas for the antipode of
NCSym in other bases? This is not yet known for the monomial basis or the homogeneous
basis.
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