aa r X i v : . [ m a t h . C O ] J a n Lessons I Learned from Richard Stanley
James Propp to Richard Stanley, on the occasion of his 70th birthday
Abstract.
I will share with the reader what I have learned from RichardStanley and the ways in which he has contributed to research in combinatoricsconducted by me and my collaborators.
1. Two big ideas
The biggest lesson I learned from Richard Stanley’s work is, combinatorial ob-jects want to be partially ordered!
By which I mean: if you are trying to understandsome class of combinatorial objects, you should look at ways of putting a partialorder on the class, in hopes of finding one that has especially nice properties. Youwon’t always succeed, but when you do, the gains are likely to more than justifythe effort.A related lesson that Stanley has taught me is, combinatorial objects wantto belong to polytopes!
That is: If you can find a way to view the objects you’reinterested in as the vertices or facets of a polytope, or as the faces (of all dimensions)of a polytope, or as the lattice points inside a polytope, then geometrical methodswill give you a lot of combinatorial insight.
2. Tilings and perfect matchings
The two articles of Stanley’s that had the greatest impact on my research were[ ] and [ ], which deal respectively with rhombus tilings of hexagons (Stanleycalls them plane partitions whose three-dimensional diagram fits inside a box) anddomino tilings of rectangles (Stanley, taking the dual point of view, calls themdimer covers).Figure 1(a) shows one of the 20 ways to tile a regular hexagon of side-length 2using twelve unit-rhombus tiles; Figure 1(b) shows the associated perfect matchingof the graph whose edges correspond to allowed positions of the tiles, with verticescorresponding to triangular “half-tiles”.Figure 2(a) shows one of the 36 ways to tile a square of side-length 4 using eight1-by-2 rectangular tiles (dominos); Figure 2(b) shows the associated dimer cover(or perfect matching) of the graph whose edges correspond to allowed positions ofthe tiles, with vertices corresponding to square half-tiles. Mathematics Subject Classification.
Primary . (a) (b)
Figure 1.
A lozenge tiling and its dual perfect matching.(a) (b)
Figure 2.
A domino tiling and its dual perfect matching.I was struck by the dissimilarity between formula (1) in [ ] and formula (2) in[ ]. The first of these implies that the number of ways to tile a regular hexagon ofside n with 3 n rhombuses of side length 1 (each consisting of two unit equilateraltriangles joined edge-to-edge) is n Y i =1 n Y j =1 n Y k =1 i + j + k − i + j + k − . The second formula implies that the number of ways to tile a 2 n -by-2 n square with2 n dominos (each consisting of two unit squares joined edge-to-edge) is n Y j =1 n Y k =1 (cid:18) πj n + 1 + 4 cos πk n + 1 (cid:19) . Why do we see simple rational numbers in the former and complicated trigonometricexpressions in the latter, when the two problems might seem at first to be soanalogous to one another? Pondering this question led me and others into deeperexploration of the dimer model of statistical physics (or what graph theorists callperfect matchings of graphs), and M.I.T. became a major center for research inthis field in the 1990s. My sole coauthored paper with Stanley [ ] was written ESSONS I LEARNED FROM RICHARD STANLEY 3 during this period. Trying to raise the visibility of this field, I took encouragementfrom [ ], whose success emboldened me to come up with a similar problems-listof my own [ ]. For an overview of the subject of enumeration of tilings, see [ ].For Stanley’s own introduction to the subject, co-written with Federico Ardila (aformer member of the M.I.T. Tilings Research Group), see [ ].A major tool in the study of tilings has been what are called height-functions.These are mathematical constructions that generalize a feature of tilings that youmay have already noticed: the human visual system is inclined to view Figure 1(a)as a projection of a stepped surface composed of squares seen at an oblique angle.Your eye and brain may not perform the same trick for Figure 2(b), but dominotilings are equally susceptible to being viewed as surfaces in three-dimensions, froma purely mathematical perspective. I learned about this point of view from workof Conway and Lagarias [ ] and Thurston [ ], and did some unpublished workshowing how the key ideas could be applied to a variety of combinatorial models[ ]. What is really going on is that the set of perfect matchings of a planar graphcan be endowed with a partial ordering that turns it into a distributive lattice; and,being at MIT, I was optimally situated to exploit this. The distributive latticestructure on tilings often gives the right way to “ q -ify” enumerative questions. MyPh.D. student David Wilson came up with a brilliant way to exploit the latticestructure to make it possible to efficiently sample from the uniform distribution onthe set of perfect matchings of a planar graph. This is the method of Coupling FromThe Past (or CFTP), originally developed for the study of tilings but applicablemuch more broadly (for an overview, see [ ] or Chapter 22 of [ ]).Stanley’s work on enumerating symmetry classes of plane partitions played aninteresting role in the advent of the notion of cyclic sieving. In [ ], Stanley intro-duced the idea of complementing a plane partition whose solid Young diagram fitsinside a specific box, and combined this new symmetry with other sorts of symme-try that MacMahon and others had already studied. John Stembridge, in exploringsome of Stanley’s new symmetry classes, noticed a curious relation between thesenew enumeration problems and some old ones. Specifically, he noticed that if onetook the q -enumeration of some old symmetry class, and set q = −
1, the resultwould be the number of plane partitions which in addition to belonging to thesymmetry class also were self-complementary. This is the q = − ]. Vic Reiner, Dennis Stanton, and Dennis White went on to real-ize that the q = − ]. To be brief (though somewhat at variancewith standard definitions and notation), we say that the quadruple ( S, π, p ( x ) , ζ )exhibits the cyclic sieving phenomenon (with S a finite set, π a permutation of S , p ( x ) a polynomial with integer coefficients, and ζ a root of unity) when for allintegers k , the number of fixed points of π k equals | p ( ζ k ) | . For recent discussionsof cyclic sieving, see [ ] and [ ].
3. Combinatorial reciprocity
I have also been inspired by Stanley’s work on combinatorial reciprocity, asdescribed in [ ] and [ ]. (This aspect of Stanley’s work was also the subject ofmy presentation at the 2004 Stanley Conference [ ].) The key result of the formerpaper is that the chromatic polynomial of a graph G , evaluated at −
1, equals( − | V ( G ) | times the number of acyclic orientations of G . When I first encountered JAMES PROPP this fact, it seemed miraculous. In what sense do the acyclic orientation of G correspond to colorings with − ] has taught us that thecombinatorial Euler measure of R is − R k is ( − k ), so the precedingsentence is not entirely strange. If one views an R -coloring of G as a point in R | V ( G ) | , then the set of R -colorings of G becomes the complement of a hyperplanearrangement, and we can view it as a disjoint union of | V ( G ) | -dimensional opencells (each homeomorphic to R | V ( G ) | ), which are in natural bijection with the acyclicorientations of G . (For a related viewpoint, see [ ].)[ ] describes many other examples in which one starts with some polynomial p ( t ) for which p ( n ) has some enumerative significance when n is a positive integer,and finds that the values of p ( n ) when n is a negative integer possess (up to sign)some sort of enumerative significance as well, reminiscent of but different from theenumerative significance of p ( n ) when n is positive. My favorite example comesfrom Ehrhart theory: If Π is a compact convex polytope, and p ( t ) is its Ehrhartpolynomial, so that p ( n ) is the number of lattice points in the n th dilation of Π(for all n = 1), then p ( − n ) is the number of lattice points in the interior of the n thdilation of Π.One can even apply reciprocity to domino tilings. For fixed k , the numberof domino tilings of a k -by- n rectangle (call it T k ( n )) satisfies a linear recurrencerelation in n that allows us to extend it to all integers n , regardless of sign. It turnsout that, with this extended definition of T k , we have T k ( − − n ) = ± T k ( n ). Fora combinatorial explanation of this, see [ ]. I am convinced that there is a lot ofimportant work yet to be done in the area of combinatorial reciprocity, and I hopeto see Stanley’s articles serve as a foundation for future progress.
4. Dynamical algebraic combinatorics
The articles of Stanley’s that I’ve drawn nourishment from most recently are[ ] and [ ]. These articles fit into a growing body of work that one might call dy-namical algebraic combinatorics (a field that arguably includes within its purviewthe cyclic sieving phenomenon described earlier). The first article takes the com-binatorial operation that turns antichains of a poset into order ideals of a posetand lifts it into a piecewise-linear map between the order polytope (whose verticescorrespond to order ideals) and the chain polytope (whose vertices correspond toantichains). The second article treats, among other things, the operation of pro-motion on linear extensions of a poset. Here I will mention a link between the twoarticles, discussed in greater length in [ ]. Sch¨utzenberger’s promotion operator onthe set of semistandard Young tableaux of rectangular shape with A rows and B columns having entries between 1 and n is naturally conjugate to an action on therational points in the order polytope of [ A ] × [ n − A ] with denominator dividing B . (Here [ n ] denotes the chain of length [ n ].) The latter action, introduced in [ ],is expressible as a composition of fundamental involutions called toggles, which inthe setting of [ ] can be seen as continuous piecewise-linear maps from the orderpolytope to itself.The notion of the order polytope has caught on, but the allied notion of thechain polytope has languished in comparison: the two search terms generated 270 ESSONS I LEARNED FROM RICHARD STANLEY 5 and 46 hits respectively in scholar.google.com in June 2014. I hope the latternotion will attract more of the attention it deserves.
5. Enumerative Combinatorics, volumes 1 and 2
No discussion of Stanley’s contributions would be complete without mentionof his books [ ] and [ ]. These books are not light reading, but they are clearlywritten and loaded with useful information. A good deal of my email correspon-dence with Stanley over the past two decades consists of me asking him a questionand him informing me that the answer to my question (or some new result of mine)is in [ ]). I would estimate that over the course of my career thus far I’ve spentseveral dozen hours rediscovering things that were already in these books.In a light-hearted vein, I expressed my appreciation for these books in theform of a song that was performed at the opening day banquet of the Stanley@70conference in 2014. It’s based on the song “Guys and Dolls” by Frank Loesser, andsome of the lines were written by Noam Elkies, who also did the arrangements andconducted the performance from the piano.What’s in Inventiones ? I’ll tell you what’s in
Inventiones .Folks provin’ theorems, ’stead-a figurin’ odds to use for bettin’on the ponies.That’s what in
Inventionies !What’s in the
Intelligencer ? I’ll tell you what’s in the
Intelli-gencer .Articles on abstruse mathematical questions for which countin’plays a role in the enswer.That’s what’s in the
Intelligencer !What’s in every math joynal? I’ll tell you what’s in every mathjoynal.Combinatorics achievin’ renown as a fountain of truths bothbeautiful and etoynal.That’s what’s in every math joynal !Combinatorialists have one trusted resource;And now it’s both a physical and an “e-”source.Yes sir! Yes ma’am!When you study balls stuck in separate stalls,Then the facts that you need are in EC One.When you seek a combinatorial truth,EC One’s where you go to see if it’s so – unless it’s in Knuth.When you see a mu with a zeta or twoAnd a delta thrown in for some extra fun ...Call it odd, call it even; it’s a principle to believe inThat the source you’re consultin’ is EC One.When you see a rook that determines a hookThen the book that you’re lookin’ at’s EC One.When a theorem features a bent letter SAll curled up in distress, the theorem’s address is not hard toguess.
JAMES PROPP
When you wend a path and some elegant mathTells the number of ways that it can be done,It’s a true proposition known to every mathematicianThat the opus you’ve opened is EC One.When your lovely proof springs a leak in its roofThen the patch for your goof is in EC One.But proceed with caution. You never can tell:Maybe page eighty-six has not just the fix but your proof as well!When an exercise makes you feel not so wise’Cause for you it ain’t “EC” – forgive the pun –Call it plus, call it minus; chalk it all up to Stanley’s slyness’Cause the book that has stumped you is EC One –Or EC Two –The book that you’re readin’ is EC One!
6. Special sequences
Of particular note is Stanley’s impressive list of combinatorial incarnationsof the Catalan numbers [ ]. No other integer sequence, not even the Fibonaccisequence, has such a rich assortment of seemingly unrelated manifestations, andthe On-line Encyclopedia of Integer Sequences page for the Catalan sequence (entryA000108) is the longest in the whole OEIS database. Given the length of Stanley’sannotated list, the most time-efficient way for a researcher to find out whether someparticular Catalan-incarnation has been noticed before is not to leaf through thewhole addendum (96 pages at present count) but to ask Stanley. Unfortunately, thatwill not be a viable method forever. As Sara Billey and Bridget Tenner have pointedout, we might start trying to think now about how to automate what Richard doeswhen he fields a question about the Catalan number literature by finding waysto assign “fingerprints” to different sorts of combinatorial objects, in a way thatmight make some form of automated search possible. This problem isn’t just ofinterest to combinatorialists; one can argue that it is a natural test-bed for themuch broader enterprise of semantic search. In identifying distinguishing structuralcharacteristics of different incarnations of the Catalan objects, and finding ways torepresent these distinctions in software, we will learn lessons that can be appliedmuch more broadly to the Artificial Intelligence problem of content-based searchingin other mathematical domains.Another integer sequence that has followed Richard around over the past decades(though not as doggedly as the Catalan sequence) is the sequence 1 , , , , , . . . .In [ ], Stanley (citing earlier work of Mills, Robbins, and Rumsey) mentions that2( n ) occurs as the sum of 2 s ( T ) , where T ranges over all n -by- n alternating signmatrices and s ( T ) is the number of − T . (For a combinatorial explanation ofthis, relating the formula to domino tilings, see [ ].) Quite recently, this sequenceresurfaced in [ ] in connection with a seemingly quite different sort of problem aris-ing from Ramsey theory. Having absorbed the lesson “combinatorial objects wantto be partially ordered” from Stanley during my many years in the Boston area,it was gratifying to have a chance to return the favor; my contribution to [ ] wassuggesting the partial ordering that was one of the keys to unlocking the problem. ESSONS I LEARNED FROM RICHARD STANLEY 7
7. The culture of combinatorics
Finally, I’d like to mention a contribution that Richard has made to the math-ematical life of the Boston area that might not otherwise be recorded, namely, hisfounding and continued leadership of the Cambridge Combinatorics Coffee Club.“CCCC” (as it is called) has served as a great way for Boston-area combinatori-alists to keep up with one another’s research interests, and a way for some of usto incubate our ideas over time. I wonder whether other branches of mathematicshave similar institutions in which they freely share work-in-progress and welcomeothers to join their projects. I suspect that one reason many of us who work incombinatorics have gravitated toward the field is how friendly and uncompetitiveits practitioners are; one rarely hears about combinatorialists racing one anothertowards the solution of some hot open problem or stealing ideas from each other.No doubt the accessibility of the subject matter of combinatorics is a largefactor in the friendliness of practitioners. But a big part is played by people likeStanley, who lead by example and set the tone for the field, creating what MargaretBayer aptly described (in her remarks at the close of the Stanley@70 conference)as “a culture of cooperation and openness”.
References
1. Federico Ardila and Richard P. Stanley,
Tilings , Math. Intelligencer (2010), no. 4, 32–43, http://arxiv.org/abs/math/0501170 .2. J. H. Conway and J. C. Lagarias, Tiling with polyominoes and combinatorial group theory , J.Combin. Theory Ser. A (1990), no. 2, 183–208.3. Richard Ehrenborg and Margaret A. Readdy, On valuations, the characteristic polynomial,and complex subspace arrangements , Adv. Math. (1998), no. 1, 32–42.4. David Einstein and James Propp,
Piecewise-linear and birational toggling , http://arxiv-web3.library.cornell.edu/abs/1404.3455 .5. Noam Elkies, Greg Kuperberg, Michael Larsen, and James Propp, Alternating-sign matricesand domino tilings , J. Algebraic Combin. (1992), 111–132, 219–234.6. David A. Levin, Yuval Peres, and Elizabeth L. Wilmer, Markov chains and mixing times ,American Mathematical Society, Providence, RI, 2009, with a chapter by James G. Proppand David B. Wilson.7. Fu Liu and Richard Stanley,
A distributive lattice connected with arithmetic progressions oflength three , http://arxiv.org/abs/1312.5758 .8. James Propp, Lattice structure for orientations of graphs , http://arxiv.org/abs/math/0209005 , 1993.9. , Enumeration of matchings: problems and progress , New perspectives in algebraiccombinatorics (Berkeley, CA, 1996–97), Math. Sci. Res. Inst. Publ., vol. 38, Cambridge Univ.Press, Cambridge, 1999, Updated 2014 version at http://arxiv.org/abs/math/9904150 ,pp. 255–291.10. ,
A reciprocity theorem for domino tilings , Electron. J. Com-bin. (2001), no. 1, Research Paper 18, 9 pp. (electronic), ,.11. , Richard stanley and combinatorial reciprocity , presented as a talk at the 2004 StanleyConference; slides available at http://jamespropp.org/stanley04.pdf , 2004.12. ,
Enumeration of tilings , Handbook of Enumerative Combinatorics, CCR Press, BocaRaton, 2015, http://jamespropp.org/eot.pdf .13. James Propp and Richard Stanley,
Domino tilings with barriers , J. Combin. Theory Ser. A (1999), no. 2, 347–356.14. James Propp and David Wilson, Coupling from the past: a user’s guide , Microsurveys indiscrete probability (Princeton, NJ, 1997), DIMACS Ser. Discrete Math. Theoret. Comput.Sci., vol. 41, Amer. Math. Soc., Providence, RI, 1998, urlhttp://jamespropp.org/user.pdf,pp. 181–192.
JAMES PROPP
15. Victor Reiner, Dennis Stanton, and Dennis White,
The cyclic sieving phenomenon , J. Combin.Theory Ser. A (2004), no. 1, 17–50.16. ,
What is . . . cyclic sieving? , Notices Amer. Math. Soc. (2014), no. 2, 169–171.17. Bruce E. Sagan, The cyclic sieving phenomenon: a survey , Surveys in combinatorics 2011,London Math. Soc. Lecture Note Ser., vol. 392, Cambridge Univ. Press, Cambridge, 2011,pp. 183–233.18. Stephen Schanuel,
Negative sets have Euler characteristic and dimension , Category theory(Como, 1990), Lecture Notes in Math., vol. 1488, Springer, Berlin, 1991, pp. 379–385.19. Richard Stanley,
Acyclic orientations of graphs , Discrete Math. (1973), 171–178.20. , Combinatorial reciprocity theorems , Combinatorics, Part 2: Graph theory; founda-tions, partitions and combinatorial geometry (Proc. Adv. Study Inst., Breukelen, 1974), Math.Centrum, Amsterdam, 1974, pp. 107–118. Math. Centre Tracts, No. 56.21. ,
On dimer coverings of rectangles of fixed width , Discrete Appl. Math. (1985),no. 1, 81–87.22. , A baker’s dozen of conjectures concerning plane partitions , Combinatoire ´enum´erative(Montreal, Que., 1985/Quebec, Que., 1985), Lecture Notes in Math., vol. 1234, Springer,Berlin, 1986, pp. 285–293.23. ,
Symmetries of plane partitions , J. Combin. Theory Ser. A (1986), no. 1, 103–113,with correction in [ ].24. , Two poset polytopes , Discrete Comput. Geom. (1986), no. 1, 9–23.25. , Erratum: “Symmetries of plane partitions” , J. Combin. Theory Ser. A (1987),no. 2, 310.26. , Plane partitions: past, present, and future , Combinatorial Mathematics: Proceedingsof the Third International Conference (New York, 1985), Ann. New York Acad. Sci., vol. 555,New York Acad. Sci., New York, 1989, pp. 397–401.27. ,
Enumerative combinatorics. Vol. 2 , Cambridge Studies in Advanced Mathematics,vol. 62, Cambridge University Press, Cambridge, 1999, with a foreword by Gian-Carlo Rotaand appendix 1 by Sergey Fomin.28. ,
Promotion and evacuation , Electron. J. Combin. (2009),no. 2, Special volume in honor of Anders Bjorner, Research Paper 9, 24, .29. , Enumerative combinatorics. Volume 1 , second ed., Cambridge Studies in AdvancedMathematics, vol. 49, Cambridge University Press, Cambridge, 2012.30. ,
Catalan addendum , , May 25,2013.31. John R. Stembridge, Some hidden relations involving the ten symmetry classes of plane par-titions , J. Combin. Theory Ser. A (1994), no. 2, 372–409.32. Jessica Striker and Nathan Williams, Promotion and rowmotion , European J. Combin. (2012), no. 8, 1919–1942, http://arxiv.org/abs/1108.1172 .33. William P. Thurston, Conway’s tiling groups , Amer. Math. Monthly (1990), no. 8, 757–773.(1990), no. 8, 757–773.