Interview with Carla D. Savage

numerativeombinatorics

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A Enumerative Combinatorics and Applicationsecajournal.haifa.ac.il

ECA 2:3 (2022) Interview #S3I9https://doi.org/10.54550/ECA2022V2S3I9

Interview with Carla D. SavageToufik Mansour

Photo by Griff Bilbro

Carla Savage completed her bachelor’s degree studies at CaseWestern Reserve University in 1973 and her master’s stud-ies at the University of Illinois, Urbana-Champaign in 1975.She obtained a Ph.D. from the University of Illinois at Ur-bana–Champaign in 1977, under the supervision of David E.Muller. She is a Professor in the Department of ComputerScience at North Carolina State University. Since 2012, sheis a Fellow of the AMS. In 2019, she became a SIAM Fellow”for outstanding research in algorithms of discrete mathematicsand computer science applications, alongside exemplary serviceto mathematics.” Her research interests include Combinatorics;enumeration and structure in combinatorial families; theory ofpartitions; linear Diophantine enumeration; lattice point enu-meration; permutation statistics; the combinatorics, geometry,and number theory of lecture hall partitions. Professor Savage

has given numerous invited talks in conferences and seminars. From 2013 up to the end ofJanuary this year, she was the secretary of the American Mathematical Society.

Mansour: Professor Savage, first of all, wewould like to thank you for accepting this in-terview. Would you tell us broadly what com-binatorics is?Savage: To me, it is the study of enumerationand structure in discrete families.Mansour: What do you think about the de-velopment of the relations between combina-torics and the rest of mathematics?Savage: I have been most aware of relationsbetween combinatorics and theoretical com-puter science. Since the early seventies, theareas have grown up together, each challeng-ing the other with intriguing problems. Someof the most recognized leaders in combinatoricshave had a strong interest in computation.Mansour: What have been some of the maingoals of your research?Savage: When I turned my attention to com-binatorics, I learned of so many interestingopen questions. I was always thinking, “What

can I compute that would give insight into thesolution of a problem, that would help a math-ematician solve the problem?”

Mansour: We would like to ask you aboutyour formative years. What were your earlyexperiences with mathematics? Did that hap-pen under the influence of your family or someother people?

Savage: I went to public school in BaltimoreCounty, Maryland. We could take advancedcourses in many areas, for instance, mathe-matics, physics, and french. Amazing teach-ers. Small classes with students who were mo-tivated, competitive, fun.

Mansour: Were there specific problems thatmade you first interested in combinatorics?

Savage: I attended the 1988 SIAM Confer-ence on Discrete Mathematics in San Francisofor the chance to hear Donald Knuth speak.But I also went to Herb Wilf’s talk there onGray codes and was intrigued by some of his

The authors: Released under the CC BY-ND license (International 4.0), Published: August 27, 2021Toufik Mansour is a professor of mathematics at the University of Haifa, Israel. His email address is [email protected]

https://doi.org/10.54550/ECA2022V2S3I9

Interview with Carla D. Savage

questions. For example, is it possible to list allpartitions of an integer n, each exactly once,so that, in moving from one partition to thenext, one part of the partition increases by 1and one part decreases by 1? In working thatout, I fell in love with integer partitions. Thatchanged everything.Mansour: What was the reason you chose theUniversity of Illinois at Urbana–Champaign foryour Ph.D. and your advisor David E. Muller?Savage: My husband and I were lookingat schools with strong graduate programs inmathematics and physics. UIUC suited usboth well. I had interests in electrical engineer-ing and computer science as well as mathemat-ics, things like coding theory, switching the-ory, fault-tolerant computing, automata the-ory, formal languages, algorithms, graph the-ory, and complexity theory. David Muller wasa mathematician who thought very abstractlyabout problems in computer science, so I washappy that he was willing to take me on. Heis the “Muller” of Reed-Muller codes.Mansour: What was the problem you workedon in your thesis?Savage: I worked on the design of parallelalgorithms for graph problems. By the mid-seventies, parallel computers were already be-ing imagined, designed, and built: the IlliacIV, STARAN, CRAY. On the horizon weremachines like the Intel Hypercube, the Con-nection Machine. And there were a varietyof models of parallel computation. One ofthose models, the PRAM (parallel random-access machine) assumed that an unlimited,but finite, number of processors could work insync, with random access to a common mem-ory, just about the best you hope for. Whatspeedup could be achieved with such a model?It turns out that many problems that can besolved in polynomial time O(nk) on a sequen-tial computer can be solved in poly-logarithmicO((log n)k) time on a PRAM with a polyno-mial number of processors.

This work was coming after a decade ofgreat strides in the design of efficient sequentialgraph algorithms by John Hopcroft, RobertTarjan, and others. However, one of their mostpowerful tools - depth-first search - seems to be

inherently sequential. So other methods had tobe developed for the PRAM.Mansour: What would guide you in your re-search? A general theoretical question or aspecific problem?Savage: Usually, a specific problem. It is funto learn something new if it might help to solveyour problem.Mansour: When you are working on a prob-lem, do you feel that something is true evenbefore you have the proof?Savage: All the time, but I am often wrong.Mansour: What three results do you considerthe most influential in combinatorics duringthe last thirty years?Savage: Here are some things in my realm ofinterest:(1) Middle levels problem. Consider the lat-tice of subsets of a 2k + 1-element set orderedby inclusion. Is there a Hamiltonian cyclein the bipartite graph formed by the k- andk + 1-element subsets? So easy to state, butmany people spent a lot of time on this prob-lem without solving it. Finally in a 2016 pa-per published in Proc. London Math. Soc.,Torsten Mutze1 showed constructively that itis always possible. In the years since, progresshas been made on several related questions. Ithas led to a consolidation of ad hoc techniquesfor Gray codes and simplifications of previousalgorithms.(2) Four color theorem. More than thirty yearsago, I know, but this was happening while Iwas a graduate student at Illinois, home ofKenneth Appel and Wolfgang Haken2, and itwas pretty exciting. It influenced so much ofthe work of the past forty years, on graphminors and the structure of graphs and, cor-respondingly, on the design of efficient algo-rithms for special classes of graphs.(3) Barvinok’s algorithm3 for lattice point enu-meration. In 1994, Alexander Barvinok gavethe first polynomial-time algorithm for count-ing the number of lattice points in a convexpolyhedron in any fixed dimension d. The al-gorithm is polynomial in the dimension d andthe size of the problem. This led to the de-velopment of software for efficient lattice pointenumeration such as LattE and its successors.

1T. Mutze, Proof of the middle levels conjecture, Proc. Lond. Math. Soc. (3) 112 (2016), no. 4, 677–713.2K. Appel and W. Haken, A proof of the four color theorem, Discrete Math. 16 (1976), no. 2, 179–180.3A. Barvinok, Polynomial time algorithm for counting integral points in polyhedra when the dimension is fixed, Math. Oper.

Res. 19 (1994), 769–779.

ECA 2:3 (2022) Interview #S3I9 2


Mansour: What are the top three open ques-tions in your list?

Savage: (1) Does L(m,n) have a symmetricchain decomposition? L(m,n) is the lattice ofinteger partitions whose Ferrers diagrams fitin an m × n box, ordered by diagram inclu-sion. In 1980, Richard Stanley4 proved thatL(m,n) is rank symmetric and unimodal andconjectured that it had the stronger propertyof possessing a decomposition into symmetricchains. The question of a symmetric chain de-composition when m ≥ 5 remains open at thistime, although Kathy O’Hara5 gave an elegantcombinatorial proof of unimodality of L(m,n)in 1990.

(2) Is the Durfee polynomial real-rooted? TheDurfee polynomial is defined by Dn(x) =∑

i≥0 p(n, d)xd where p(n, d) is the number ofpartitions of n with Durfee square of size d.With Rod Canfield and Sylvie Corteel6 in 1998we showed that, as n tends to infinity, thesequence of coefficients of Dn(x) is asymp-totically normal, unimodal, and log-concave.However, empirical evidence led us to conjec-ture that Dn(x) has only real roots for all pos-itive n, a stronger property which would alsoimply that the average size and most likely sizeof the Durfee square of a partition of n differby at most 1.

(3) Do simple symmetric n-Venn diagrams ex-ist for all prime n? With Jerry Griggs andChip Killian7, we showed in 2004 that rota-tionally symmetric Venn diagrams for n sets(like the familiar one for three sets) exist forall prime n. But there may be points where3 or more of the n curves intersect. Can thesame result be achieved with simple Venn di-agrams in which at most two curves intersectat a point? This was known to be possiblefor n = 1, 2, 3, 5, 7 and more recently, thanksto Khalegh Mamakani and Frank Ruskey8, forn = 11, 13.

Mansour: Do you think that there are core ormainstream areas in mathematics? Are sometopics more important than others?

Savage: For many years the AMS Steele

Prize for Seminal Contribution to Researchwas given in five “core” areas of mathematics,with the prize being awarded in one area eachyear on a five-year cycle. A few years ago, thefive core areas were redefined to better coverresearch mathematics: Analysis/Probability,Algebra/Number Theory, Applied Mathemat-ics, Geometry/Topology, and Discrete Math-ematics/Logic. In addition, a sixth category“Open” was included, resulting in a six-yearcycle for the prize. I think this is a recognitionthat there are core areas, but that importantresearch these days may cross boundaries.

Mansour: What do you think about the dis-tinction between pure and applied mathemat-ics that some people focus on? Is it mean-ingful at all in your case? How do you see therelationship between so-called “pure” and “ap-plied” mathematics?

Savage: I am probably seen as an appliedmathematician since I have spent my wholecareer in a computer science department. Buteven within computer science, there are moretheoretical and more applied areas. Both hadimportant roles to play in the progress of thelast fifty years, thanks to researchers with thevision to put them together.

Mansour: What advice would you give toyoung people thinking about pursuing a re-search career in mathematics?

Savage: Learn what mathematics is good for.I do not think my generation did that very well.

Mansour: Would you tell us about your in-terests besides mathematics?

Savage: Family, film, books, music, nature,simple pleasures.

Mansour: You have been the Secretary ofthe American Mathematical Society for eightyears. Would you tell us about your experi-ences? How do you manage to balance yourtime among research, teaching, and your pro-fessional service?

Savage: My experience with the AMS has leftme in awe of the tremendous service mathe-maticians are willing to do for our field. Thereare hundreds of volunteers appointed each year

4R. P. Stanley, Weyl groups, the hard Lefschetz theorem, and the Sperner property, SIAM J. Algebraic and Discrete Methods1 (1980), 168–184.

5K. O’Hara, Unimodality of Gaussian coefficients: a constructive proof, J. Comb. Theory, Ser. A 53 (1990), 29–52.6E.R. Canfield, S. Corteel, and C.D. Savage, Durfee polynomials, Elec. J. Comb. 5 (1998), R32.7J. Griggs, C. E. Killian, and C. D. Savage, Venn diagrams and symmetric chain decompositions in the boolean lattice, Elec.

J. Comb. 11:1 (2004), #R2.8K. Mamakani and F. Ruskey, New roses: simple symmetric Venn diagrams with 11 and 13 curves, Discrete Comput. Geom.

52:1 (2014), 71–87.



to editorial and other AMS committees. Ithink it is a challenge for all of us to balanceour time between research, teaching, and pro-fessional service. As AMS secretary, I was in-volved in some way with most aspects of thesociety: publications, conferences, prizes, com-mittees, elections. I was in a unique positionto see how the pieces fit together and to workbehind the scenes to help make things happen.I am grateful to have had that opportunity.Mansour: You have written an excellent sur-vey paper on Gray codes titled as A surveyof combinatorial Gray codes9. Would you tellus about Gray codes and their combinatorialaspects? There is also a connection betweenGray codes and music theory. Would you ex-plore more about this connection?Savage: A Gray code for a combinatorial fam-ily is a listing of the objects in the family, eachexactly once, in such a way that successive ob-jects differ only in a prescribed (usually small)way. For example, the binary reflected Graycode lists all n-bit strings so that successivestrings differ in only one-bit position. A Graycode for permutations might list all permuta-tions of {1, . . . , n} so that successive permuta-tions differ only by a transposition; there aremany other conditions on successive elementsthat one might prescribe. A permutation Graycode can be interpreted as a score for ringingn church bells in n! rounds, where each permu-tation specifies the order in which to ring then bells in that round.

A Gray code can be viewed as a Hamilto-nian path in the graph whose vertices are theobjects in the combinatorial family and whereobjects are adjacent if they differ by the pre-scribed change.

Gray code schemes typically involve decom-posing a combinatorial structure into smallersubstructures which are then listed recursively.How to combine the sublists to enforce theGray code restriction at the boundaries is thehard part. Frank Ruskey and his former stu-dents Joe Sawada and Aaron Williams havebeen responsible for (and continue to produce)some of the most delicately clever Gray codeconstructions. Donald Knuth included severalresults and open questions about Gray codesin Volume 4A of The Art of Computer Pro-

gramming.Mansour: In a very recent short arti-cle, published at Newsletter of the Euro-pean Mathematical Society, Professor MelvynB. Nathanson10, while elaborating on theethical aspects of the question “Who Ownsthe Theorem?” concluded that “Mathematicaltruths exist, and mathematicians only discoverthem.” On the other hand, some people claimthat “mathematical truths are invented,” andsome others claim that it is both invented anddiscovered. What do you think about this olddiscussion?Savage: I found Nathanson’s article quite in-teresting, especially the ethical issues raised.It reminds me of questions about whether al-gorithms can be patented - in most cases, theycannot be, they are considered abstract ideas.But the software implementation of the algo-rithm might be.

I think mathematical truths are discovered.But perhaps, sometimes the questions thatlead to their discovery are invented.Mansour: Lecture Hall partitions has recentlybecome an active area of research and you havesome interesting works in this direction. Whyare they important? Do they have some sur-prising connections with other combinatorialobjects?Savage: Lecture hall partitions Ln, intro-duced by Mireille Bousquet-Melou and KimmoEriksson11 in 1997, are integer sequences λ =λ1, . . . , λn satisfying

0 ≤ λ1

1≤ λ2

2≤ . . . ≤ λn

n.

Bousquet-Melou and Eriksson showed that∑λ∈Ln

q∑n

i=1 λi =1∏n

j=1(1− q2j−1),

a result known as the lecture hall theorem. Itis surprising that these partitions, defined bythe ratio of consecutive parts, would have sucha simple generating function. Moreover, theright-hand side is the generating function forpartitions into odd parts less than 2n. In fact,the lecture hall theorem is a new finite versionof Euler’s theorem that the number of parti-tions of an integer into distinct parts is thesame as the number of its partitions into odd

9C. D. Savage, A survey of combinatorial Gray codes, SIAM Rev. 39:4 (1997), 605–629.10See https://www.ems-ph.org/journals/newsletter/pdf/2020-12-118.pdf.11M. Bousquet-Melou and K. Eriksson, Lecture hall partitions, Ramanujan J. 1:1 (1997), 101–111.


https://www.ems-ph.org/journals/newsletter/pdf/2020-12-118.pdf


parts. Bousquet-Melou and Eriksson12 also de-

fined s-lecture hall partitions L(s)n for nonneg-

ative integer sequences s as sequences λ satis-fying

0 ≤ λ1

s1

≤ λ2

s2

≤ . . . ≤ λnsn

and looked for sequences s that would yield in-teresting enumerative results. As one example,they showed that if s is defined, for positiveinteger `, by sn = `sn−1 − sn−2 with s0 = 0,s1 = 1, then∑

λ∈L(s)n

q∑n

i=1 λi =1∏n

j=1(1− qsj−1+sj).

Since then, many others have looked at aspectsof lecture hall partitions. As I wrote in my2016 survey paper, The Mathematics of Lec-ture Hall Partitions13, over the past 25 yearslecture hall partitions have emerged as fun-damental structures in combinatorics, numbertheory, algebra, and geometry, leading to newgeneralizations and interpretations of classicaltheorems and new results.Mansour: In joint work with Michael J.Schuster, Ehrhart series of lecture hall poly-topes and Eulerian polynomials for inversionsequences14, you introduced s-lecture hall poly-topes, s-inversion sequences and studied rele-vant statistics on both families. What was themain motivation behind their studies?Savage: We found earlier15 that the numberof lecture hall partitions λ with λn/n ≤ t is(t+ 1)n. This suggests a connection with per-mutations since the Eulerian polynomial En(x)satisfies∑t≥0

(t+ 1)nxt =

∑π∈Sn

xdes(π)

(1− x)n+1=

En(x)

(1− x)n+1.

I was first able to explain the connection bijec-tively in a paper with Katie Bright16 in 2010.In 2012, with Michael Schuster, we were look-ing for a generalization for s-lecture hall par-titions, but that required a way to general-

ize permutations. Mike’s idea was to use s-

inversion sequences I(s)n , defined as

I(s)n = {e ∈ Zn | 0 ≤ ei < si, i = 1, . . . , n}.

A key ingredient to make things work was to

define the ascent statistic over I(s)n in the right

way. Position i is an ascent in s-inversion se-quence e if ei/si < ei+1/si+1. And position 0is an ascent if 0 < e1. With this definitionwe were able to show bijectively that if i

(s)n (t)

is the number of s-lecture hall partitions withλn/sn ≤ t, then∑

t≥0

i(s)n (t)xt =

∑e∈I(s)n

xasc(e)

(1− x)n+1.

We called the numerator of the right-hand side

the s-Eulerian polynomial E(s)n (x) and defined

the s-lecture polytope P(s)n as the set of all real

points λ ∈ Rn satisfying

0 ≤ λ1

s1

≤ λ2

s2

≤ . . . ≤ λnsn≤ 1.

Then i(s)n (t) is the Ehrhart polynomial of the

s-lecture hall polytope. And E(s)n is the h∗-

polynomial of the s-lecture hall polytope, aswell as the ascent polynomial of the s-inversionsequences. Following the bijection allowed usto define and track several other meaningfulstatistics.Mansour: In joint work with Mirko Vison-tai, The s-Eulerian polynomials have only realroots17, you proved a conjecture of Brenti,showing that Eulerian polynomials for all finiteCoxeter groups have only real roots. Therein,you also partially settled a conjecture of Dilks,Petersen, and Stembridge on type-B affine Eu-lerian polynomials. Further, by extending re-sults to q-analogs, you have shown that theMacMahon–Carlitz q-Eulerian polynomial hasonly real roots whenever q is a positive realnumber, thus confirming a conjecture of Chowand Gessel. In the end, you provided the argu-ments of the validity of the results for the case

12M. Bousquet-M elou and K. Eriksson, Lecture hall partitions. II, Ramanujan J. 1(2) (1997), 165–185.13C. D. Savage, The mathematics of lecture hall partitions, J. Combin. Theory Ser. A 144 (2016), 443–475.14C. D. Savage and M. J. Schuster, Ehrhart series of lecture hall polytopes and Eulerian polynomials for inversion sequences,

J. Combin. Theory Ser. A 119:4 (2012), 850–870.15S. Corteel, S. Lee, and C. D. Savage, Enumeration of sequences constrained by the ratio of consecutive parts, S em. Lothar.

Combin. 54A (2005/07), Article B54Aa.16K. L. Bright and C. D. Savage, The geometry of lecture hall partitions and quadratic permutation statistics, In 22nd Inter-

national Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010), Discrete Math. Theor. Comput. Sci.Proc., AN, pages 569–580. Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 2010.

17C. D. Savage and M. Visontai, The s-Eulerian polynomials have only real roots, Trans. Amer. Math. Soc. 367:2 (2015),1441–1466.



of the hyperoctahedral group and the wreathproduct groups, confirming further conjecturesof Chow and Gessel, and Chow and Mansour,respectively. Would you tell us about the im-portant ideas behind the proofs?

Savage: Analogous to the way the Eulerianpolynomial is viewed as the descent polyno-mial of permutations, the s-Eulerian polyno-

mials E(s)n (x) are the ascent polynomials of

the s-inversion sequences. There is a bijec-tion between inversion sequences and permu-tations that sends ascents to descents. Simi-larly, we showed there is a bijection between(2, 4, . . . , 2n)-inversion sequences and signedpermutations that send ascents to descents. Sothe s-Eulerian polynomials for s = (1, 2, . . . , n)and s = (2, 4, . . . , 2n) are, respectively, thetype-A and type-B Eulerian polynomials.

Visontai and I showed that all s-Eulerianpolynomials, and therefore type-A and type-B, have only real roots. There were two keyideas involved in the proof. The first was to

use a refinement of E(s)n (x), natural for inver-

sion sequences, that could be defined recur-sively as a sum of smaller such polynomials,some of which are multiplied by x. The sec-ond idea was to use the method of “compatiblepolynomials” of Maria Chudnovsky and PaulSeymour18 to show that real rootedness of thesum followed from the real-rootedness of thesmaller polynomials as long as the factor of xcame in at just the right place (which, luckily,it does). In a section of his paper Unimodality,log-concavity, real-rootedness and beyond, Pet-ter Branden19 has given a nice explanation ofour method and generalized it to apply morebroadly.

Mansour: What kind of mathematics wouldyou like to see in the next ten-to-twenty yearsas the continuation of your work?

Savage: I think there is more to learn aboutlecture hall partitions. Here are a few exam-ples below of directions since my 2016 surveypaper.

Sylvie Corteel and Jang Soo Kim20 recently

discovered a two-dimensional version, lecturehall tableaux, with a surprisingly simple prod-uct form generating function. The generat-ing function arises as the moment of certainorthogonal polynomials that Corteel and Kimwere studying.

Petter Branden and Madeleine Leander21

have a recent paper on lecture hall P -partitions. I think Branden and Leander’smethods could be used to derive Corteeland Kim’s formula for bounded lecture halltableaux, but I haven’t figured out how yet.

Branden and Liam Solus22 have a recent pa-per on the algebraic properties of lecture hallpolytopes.

Also, the s-inversion sequences can beviewed as a combinatorial generalization ofpermutations. They can be used to interpret,refine, and generalize formulas in the same waythat structures such as partitions and permu-tations might be used. For example, Mirko Vi-sontai17 and I used them to interpret the type-A and type-B Eulerian polynomials. More re-cently, they are being used to re-interpret andgeneralize results on pattern avoiding permu-tations.Mansour: In a very recent pre-print, Coef-ficients of the inflated Eulerian polynomials23,co-authored with Juan S. Auli and Ron Gra-ham, you proved a conjecture of Pensyl andSavage by showing that the inflated s-Eulerianpolynomials are unimodal for all choices of pos-itive integer sequences s. Would you elaborateon this work and possible future research di-rections?Savage: In the same way that s-Eulerian poly-nomials arise as h∗-polynomials of s-lecture

hall polytopes P(s)n , the inflated s-Eulerian

polynomials arise from the rational s-lecture

hall polytopes, Q(s)n , consisting of all points

λ ∈ P(s)n with λn ≤ 1. Thomas Pensyl and

I showed that the inflated s-Eulerian polyno-mial24 has the explicit expression

Q(s)n (x) =

∑e=(e1,...,en)∈I(s)n

xsnasc(e)−en .

18M. Chudnovsky and P. Seymour, The roots of the independence polynomial of a clawfree graph, J. Combin. Theory Ser. B97:3 (2007), 350–357.

19P. Branden, Unimodality, log-concavity, real-rootedness and beyond, Handbook of enumerative combinatorics, 437–483.20S. Corteel and J. S. Kim, Lecture hall tableaux, Adv. Math. 371 (2020), 107266.21P. Branden and M. Leander, Lecture hall P -partitions, J. Comb. 11:2 (2020), 391–412.22P. Branden and L. Solus, Some algebraic properties of lecture hall polytopes, Sem. Lothar. Combin. 84B (2020), Article 25.23J. S. Auli, R. Graham, and C. D. Savage, Coefficients of the inflated Eulerian polynomial, arXiv:1504.01089.24T. W. Pensyl and C. D. Savage, Rational lecture hall polytopes and inflated Eulerian polynomials, Ramanujan J. 31 (2013),

97–114.



The coefficients of Q(s)n (x) seemed to form a

unimodal sequence for all s, but we could notprove it.

Later, I saw similar polynomials

Tn(x) =∑π∈Sn

xn des(π)−(n−πn)

appearing in a paper by Fan Chung and RonGraham25. One of the things Chung and Gra-ham proved was that Tn(x)/(1 + x + · · · +xn−1) and Tn−1(x) have the same sequences ofnonzero coefficients. Juan Auli and I foundthat this same property held for the inflateds-Eulerian polynomials when s is nondecreas-ing and we posted the result to the arXiv.Ron saw our preprint and said he thought hecould prove the coefficients were unimodal, sowe joined forces, and the paper you mentionedcame to be.Mansour: It is a fact that not many womenfollow a professional career in mathematics. Itis a discussion around the globe on how toget more women into mathematics. What doyou think about this issue? How do you com-pare the working conditions for women whenyou started your career and nowadays? Whatshould be done in the next ten years to involvemore women in mathematics?Savage: As an undergraduate, I went to an en-gineering school where there were only a hand-ful of women. All of the students, male andfemale, struggled through the same freshmancourses. The feeling of a shared experiencestayed with me through graduate school andas a faculty member in a relatively new depart-ment trying to build its research programs.

However, I do not think I would havehad a career in mathematics research with-out the kindness and encouragement of othermathematicians. I am especially grateful toHerb Wilf, Frank Ruskey, George Andrews,Sylvie Corteel, Donald Knuth, Fan Chung, andRichard Stanley for support, advice, and some-times just saying the right thing at the righttime. And there are so many others, may Ikeep going? My point here is that we all have achance to make a difference - to make a female(or any) mathematician feel welcome, appreci-ated, respected.

Mansour: You are a fellow of AMS andSIAM. Election as a fellow is an honor be-stowed upon members by their peers as dis-tinguished for their contributions to the disci-pline. What do you think about the impor-tance of being recognized by your fellows?Savage: It is such a great honor to me. Therecognition carries a lot of weight within mycollege. At the same time, it is humbling sinceI am well aware of other mathematicians whoare at least as deserving.Mansour: In your work, you have extensivelyused combinatorial reasoning to address im-portant problems. How do enumerative tech-niques engage in your research?Savage: I almost always start out by writing alittle program to generate or count something.If an enumerative result emerges, I rely on ba-sic techniques like recursion, generating func-tions, q-series identities, bijections to prove it.If that is not good enough, I try to learn what-ever is required.Mansour: Would you tell us about yourthought process for the proof of one of yourfavorite results? How did you become inter-ested in that problem? How long did it takeyou to figure out a proof? Did you have a “eu-reka moment”?Savage: I am more likely to have a eurekamoment when I see a pretty answer emerge - itlets you know you are finally asking the rightquestion. Once you “see” the answer, provingit can be a challenge and require you to learnnew things.

For example, we wondered how many lec-ture hall partitions15 had Ferrers diagrams thatwould fit in an n×nt box. The answer, (t+1)n,was quite a surprise and led to the discoveryof the connections with Ehrhart theory.

As another example, if, instead of lecturehall partitions, you enumerate anti-lecture hallcompositions An, i.e., integer sequences λ sat-isfying

0 ≤ λ1/n ≤ λ2/(n− 1) ≤ . . . ≤ λn/1

you find that26

∑λ∈An

q∑n

i=1 λi =n∏i=1

1 + qi

1− qi+1.

25F. Chung and R. Graham, Inversion-descent polynomials for restricted permutations, J. Combin. Theory Ser. A 120:2 (2013),366–378.

26S. Corteel and C. D. Savage, Anti-lecture hall compositions, Discrete Math. 263(1-3) (2003), 275–280.



That was another surprise that led us to dis-cover connections with other q-series identities.

I have gotten “aha” moments during IraGessel’s talks. He explains things so clearlythat I realize I can use his methods for one ofthe problems I am working on.Mansour: Is there a specific problem youhave been working on for many years? Whatprogress have you made?Savage: I return to the question of whetheror not the Durfee polynomials Dn(x) =∑

d p(n, d)xd are real-rooted whenever I learnsomething that might apply. But maybe theirmost interesting property is already settled.In the 1998 paper with Canfield and Corteel27

where we showed that the coefficients of Dn(x)

are asymptotically normal as n → ∞, we alsoshowed that the most likely size of the Dur-fee square for a partition of n is asymptotic to√

6 log 2π

√n. This means (as Alexander Yong28

pointed out in his 2014 Notices article) thatyour h-index is very likely to be about 0.54times the square root of your total number ofcitations.

Mansour: Professor Carla Savage, I wouldlike to thank you for this very interesting in-terview on behalf of the journal EnumerativeCombinatorics and Applications.

Savage: Thank you, Professor Mansour, forthe opportunity and for such interesting ques-tions.

27E.R. Canfield, S. Corteel, and C.D. Savage, Durfee polynomials, Elec. J. Comb. 5 (1998), R32.28A. Yong, Critique of Hirsch’s Citation Index: a combinatorial Fermi problem, Notices Amer. Math. Soc. 61 #9 (2014),

1040-1050.


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Interview with Carla D. Savage