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Simplicial and cellular trees

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Recent Trends in Combinatorics

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 159))

Abstract

Much information about a graph can be obtained by studying its spanning trees. On the other hand, a graph can be regarded as a 1-dimensional cell complex, raising the question of developing a theory of trees in higher dimension. As observed first by Bolker, Kalai, and Adin, and more recently by numerous authors, the fundamental topological properties of a tree — namely acyclicity and connectedness — can be generalized to arbitrary dimension as the vanishing of certain cellular homology groups. This point of view is consistent with the matroid-theoretic approach to graphs, and yields higher-dimensional analogues of classical enumerative results including Cayley’s formula and the matrix-tree theorem. A subtlety of the higher-dimensional case is that enumeration must account for the possibility of torsion homology in trees, which is always trivial for graphs. Cellular trees are the starting point for further high-dimensional extensions of concepts from algebraic graph theory including the critical group, cut and flow spaces, and discrete dynamical systems such as the abelian sandpile model.

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Notes

  1. 1.

    This weighted boundary was defined as ∂ k D k in [44]. Here we adopt the equivalent convention of [25] in order to eliminate squares on the indeterminates in weighted enumerators.

  2. 2.

    There is a slight notational error in [47], where the L Γ in Prop. 3.5 should refer to the reduced Laplacian formed by removing the rows and columns corresponding to cells of Γ.

  3. 3.

    One needs the convention that w ∅ = 1, so that L −1 alg(Q n ) is just the 1 × 1 matrix with entry \(\sum _{v\in V (Q_{n})}w_{v} =\prod _{ j=1}^{n}(x_{j} + y_{j})\).

  4. 4.

    In [44], it was erroneously stated that \(L_{k}^{\mathrm{ud}}(X) = L_{d-k}^{\mathrm{ud}}(Y )\). In fact, duality interchanges up-down and down-up Laplacians.

References

  1. G. Aalipour, A.M. Duval, Weighted spanning tree enumerators of color-shifted complexes, in preparation

    Google Scholar 

  2. G. Aalipour, A.M. Duval, J.L. Martin, A weighted cellular matrix-tree theorem, with applications to complete colorful and cubical complexes, arXiv:1510.00033[math.CO]

  3. R.M. Adin, Counting colorful multi-dimensional trees. Combinatorica 12(3), 247–260 (1992) [ ​MR 1195888 (93j:05036)]

    Google Scholar 

  4. T.L. Austin, The enumeration of point labelled chromatic graphs and trees. Can. J. Math. 12, 535–545 (1960) [ ​MR 0139544 (25#2976)]

    Google Scholar 

  5. E. Babson, I. Novik, Face numbers and nongeneric initial ideals. Electron. J. Comb. 11(2), Research Paper 25, 23 pp. (electronic) (2004/2006) [ ​MR 2195431 (2007c:05202)]

    Google Scholar 

  6. R. Bacher, P. de la Harpe, T. Nagnibeda, The lattice of integral flows and the lattice of integral cuts on a finite graph. Bull. Soc. Math. France 125(2), 167–198 (1997) [​ ​MR 1478029(99c:05111)]

    Google Scholar 

  7. C. Bajo, B. Burdick, S. Chmutov, On the Tutte-Krushkal-Renardy polynomial for cell complexes. J. Comb. Theory Ser. A 123, 186–201 (2014) [ ​MR 3157807]

    Google Scholar 

  8. P. Bak, C. Tang, K. Wiesenfeld, Self-organized criticality. Phys. Rev. A (3) 38(1), 364–374 (1988) [ ​MR 949160 (89g:58126)]

    Google Scholar 

  9. M. Baker, S. Norine, Riemann-Roch and Abel-Jacobi theory on a finite graph. Adv. Math. 215(2), 766–788 (2007) [ ​MR 2355607 (2008m:05167)]

    Google Scholar 

  10. M. Baker, F. Shokrieh, Chip-firing games, potential theory on graphs, and spanning trees. J. Comb. Theory Ser. A 120(1), 164–182 (2013) [ ​MR 2971705]

    Google Scholar 

  11. M. Beck, S. Hoşten, Cyclotomic polytopes and growth series of cyclotomic lattices. Math. Res. Lett. 13(4), 607–622 (2006) [ ​MR 2250495 (2007h:52018)]

    Google Scholar 

  12. M. Beck, F. Breuer, L. Godkin, J.L. Martin, Enumerating colorings, tensions and flows in cell complexes. J. Comb. Theory Ser. A 122, 82–106 (2014) [ ​MR 3127679]

    Google Scholar 

  13. L.W. Beineke, R.E. Pippert, Properties and characterizations of k-trees. Mathematika 18 141–151 (1971) [ ​MR 0288046 (44 #5244)]

    Google Scholar 

  14. B. Benson, D. Chakrabarty, P. Tetali, G-parking functions, acyclic orientations and spanning trees. Discret. Math. 310(8), 1340–1353 (2010) [ ​MR 2592488 (2011i:05152)]

    Google Scholar 

  15. O. Bernardi, On the spanning trees of the hypercube and other products of graphs. Electron. J. Comb. 19(4), Paper 51, 16 pp. (2012) [ ​MR 3007186]

    Google Scholar 

  16. O. Bernardi, C.J. Klivans, Directed rooted forests in higher dimension, arXiv:1512.07757 [math.CO]

  17. N.L. Biggs, Chip-firing and the critical group of a graph. J. Algebraic Comb. 9(1), 25–45 (1999) [ ​MR 1676732 (2000h:05103)]

    Google Scholar 

  18. N. Biggs, P. Winkler, Chip-firing and the chromatic polynomial. Technical Report LSE-CDAM-97-03, London School of Economics, Center for Discrete and Applicable Mathematics, 1997

    Google Scholar 

  19. A. Björner, G. Kalai, An extended Euler-Poincaré theorem. Acta Math. 161(3-4), 279–303 (1988) [ ​MR 971798 (89m:52009)]

    Google Scholar 

  20. A. Björner, L. Lovász, P.W. Shor, Chip-firing games on graphs. European J. Comb. 12(4), 283–291 (1991) [ ​MR 1120415 (92g:90193)]

    Google Scholar 

  21. E.D. Bolker, Simplicial geometry and transportation polytopes. Trans. Am. Math. Soc. 217 121–142 (1976) [ ​MR 0411983 (54 #112)]

    Google Scholar 

  22. C.W. Borchardt, Über eine Interpolationsformel für eine Art Symmetrischer Functionen und über Deren Anwendung. Math. Abh. der Akademie der Wissenschaften zu Berlin 1–20 (1860)

    Google Scholar 

  23. T. Brylawski, J. Oxley, The Tutte polynomial and its applications, Matroid applications, Encyclopedia of Mathematics and its Applications, vol. 40 (Cambridge University Press, Cambridge, 1992), pp. 123–225 [ ​MR 1165543 (93k:05060)]

    Google Scholar 

  24. Y. Burman, A. Ploskonosov, A. Trofimova, Matrix-tree theorems and discrete path integration. Linear Algebra Appl. 466, 64–82 (2015) [ ​MR 3278240]

    Google Scholar 

  25. M.J. Catanzaro, V.Y. Chernyak, J.R. Klein, Kirchhoff’s theorems in higher dimensions and Reidemeister torsion. Homology Homotopy Appl. 17(1), 165–189 (2015) [ ​MR 3338546]

    Google Scholar 

  26. D. Chebikin, P. Pylyavskyy, A family of bijections between G-parking functions and spanning trees. J. Comb. Theory Ser. A 110(1), 31–41 (2005) [ ​MR 2128964 (2005m:05010)]

    Google Scholar 

  27. F.R.K. Chung, Spectral Graph Theory, in CBMS Regional Conference Series in Mathematics, vol. 92. Published for the Conference Board of the Mathematical Sciences, Washington, DC (American Mathematical Society, Providence, RI, 1997) [ ​MR 1421568 (97k:58183)]

    Google Scholar 

  28. R. Cordovil, B. Lindström, Simplicial Matroids, Combinatorial geometries, Encyclopedia of Mathematical and its Application, vol. 29 (Cambridge Univ. Press, Cambridge, 1987), pp. 98–113 [ ​MR 921070]

    Google Scholar 

  29. R. Cori, Y. Le Borgne, The sand-pile model and Tutte polynomials. Adv. Appl. Math. 30 (1-2), 44–52 (2003), Formal power series and algebraic combinatorics (Scottsdale, AZ, 2001) [ ​MR 1979782 (2004d:05095)]

    Google Scholar 

  30. H.H. Crapo, A higher invariant for matroids. J. Comb. Theory 2, 406–417 (1967) [ ​MR 0215744 (35 #6579)]

    Google Scholar 

  31. D.M. Cvetković, The spectral method for determining the number of trees. Publ. Inst. Math. (Beograd) (N.S.) 11(25), 135–141 (1971) [ ​MR 0309772 (46 #8877)]

    Google Scholar 

  32. D.M. Cvetković, M. Doob, H. Sachs, Spectra of Graphs: Theory and Application. Pure and Applied Mathematics, vol. 87 (Academic, New York-London, 1980) [Harcourt Brace Jovanovich, Publishers] [ ​MR 572262 (81i:05054)]

    Google Scholar 

  33. M. D’Adderio, L. Moci, Ehrhart polynomial and arithmetic Tutte polynomial. Eur. J. Comb. 33(7), 1479–1483 (2012) [ ​MR 2923464]

    Google Scholar 

  34. M. D’Adderio, L. Moci, Arithmetic matroids, the Tutte polynomial and toric arrangements. Adv. Math. 232, 335–367 (2013) [ ​MR 2989987]

    Google Scholar 

  35. M. D’Adderio, L. Moci, Graph colorings, flows and arithmetic Tutte polynomial. J. Comb. Theory Ser. A 120(1), 11–27 (2013) [ ​MR 2971693]

    Google Scholar 

  36. G. Denham, The combinatorial Laplacian of the Tutte complex. J. Algebra 242(1), 160–175 (2001) [ ​MR 1844702 (2002h:05039)]

    Google Scholar 

  37. A.K. Dewdney, Higher-dimensional tree structures. J. Combinatorial Theory Ser. B 17, 160–169 (1974) [ ​MR 0369115 (51 #5351)]

    Google Scholar 

  38. D. Dhar, Self-organized critical state of sandpile automaton models. Phys. Rev. Lett. 64(14), 1613–1616 (1990) [ ​MR 1044086 (90m:82053)]

    Google Scholar 

  39. J. Dodziuk, V.K. Patodi, Riemannian structures and triangulations of manifolds. J. Indian Math. Soc. (N.S.) 40(1-4), 1–52 (1977/1976) [ ​MR 0488179 (58 #7742)]

    Google Scholar 

  40. A.M. Duval, A combinatorial decomposition of simplicial complexes. Israel J. Math. 87(1-3), 77–87 (1994) [ ​MR 1286816 (96e:52023)]

    Google Scholar 

  41. A.M. Duval, V. Reiner, Shifted simplicial complexes are Laplacian integral. Trans. Am. Math. Soc. 354(11), 4313–4344 (2002) [ ​MR 1926878 (2003j:15017)]

    Google Scholar 

  42. A.M. Duval, P. Zhang, Iterated homology and decompositions of simplicial complexes. Israel J. Math. 121, 313–331 (2001) [ ​MR 1818393 (2003a:52013)]

    Google Scholar 

  43. A.M. Duval, C.J. Klivans, J.L. Martin, Simplicial matrix-tree theorems. Trans. Am. Math. Soc. 361(11), 6073–6114 (2009) [ ​MR 2529925 (2011a:05385)]

    Google Scholar 

  44. A.M. Duval, C.J. Klivans, J.L. Martin, Cellular spanning trees and Laplacians of cubical complexes. Adv. Appl. Math. 46(1-4), 247–274 (2011) [ ​MR 2794024 (2012e:05182)]

    Google Scholar 

  45. A.M. Duval, C.J. Klivans, J.L. Martin, Critical groups of simplicial complexes. Ann. Comb. 17(1), 53–70 (2013) [ ​MR 3027573]

    Google Scholar 

  46. A.M. Duval, B. Goeckner, C.J. Klivans, J.L. Martin, A non-partitionable Cohen-Macaulay simplicial complex, arXiv:1504.04279 (2015, preprint)

  47. A.M. Duval, C.J. Klivans, J.L. Martin, Cuts and flows of cell complexes. J. Algebraic Combinatorics 41, 969–999 (2015) [ ​MR 3342708]

    Google Scholar 

  48. B. Eckmann, Harmonische Funktionen und Randwertaufgaben in einem Komplex. Comment. Math. Helv. 17, 240–255 (1945) [ ​MR 0013318 (7,138f)]

    Google Scholar 

  49. R. Ehrenborg, S. van Willigenburg, Enumerative properties of Ferrers graphs. Discret. Comput. Geom. 32(4), 481–492 (2004) [ ​MR 2096744 (2005j:05076)]

    Google Scholar 

  50. S. Faridi, The facet ideal of a simplicial complex. Manuscripta Math. 109(2), 159–174 (2002) [ ​MR 1935027 (2003k:13027)]

    Google Scholar 

  51. M. Fiedler, J. Sedláček, Über Wurzelbasen von gerichteten Graphen. Časopis Pěst. Mat. 83, 214–225 (1958) [ ​MR 0097071 (20 #3551)]

    Google Scholar 

  52. A. Fink, L. Moci, Matroids over a ring. J. Eur. Math. Soc., arXiv:1209.6571, (to appear, 2012)

  53. R. Forman, Morse theory for cell complexes. Adv. Math. 134(1), 90–145 (1998) [ ​MR 1612391 (99b:57050)]

    Google Scholar 

  54. R. Forman, A user’s guide to discrete Morse theory. Sém. Lothar. Combin. 48, Art. B48c, 35 (2002) [ ​MR 1939695 (2003j:57040)]

    Google Scholar 

  55. J. Friedman, Computing Betti numbers via combinatorial Laplacians. Algorithmica 21(4), 331–346 (1998) [ ​MR 1622290 (99c:52022)]

    Google Scholar 

  56. J. Friedman, P. Hanlon, On the Betti numbers of chessboard complexes. J. Algebraic Combinatorics 8(2), 193–203 (1998) [ ​MR 1648484 (2000c:05155)]

    Google Scholar 

  57. C. Godsil, G. Royle, Algebraic Graph Theory, Graduate Texts in Mathematics, vol. 207 (Springer, New York, 2001) [ ​MR 1829620 (2002f:05002)]

    Book  MATH  Google Scholar 

  58. I. Gorodezky, I. Pak, Generalized loop-erased random walks and approximate reachability. Random Struct. Algoritm. 44(2), 201–223 (2014) [ ​MR 3158629]

    Google Scholar 

  59. J. Guzmán, C. Klivans, Chip-firing and energy minimization on M-matrices. J. Comb. Theory Ser. A 132, 14–31 (2015) [ ​MR 3311336]

    Google Scholar 

  60. J. Guzman, C. Klivans, Chip-firing on general invertible matrices, arXiv:1508.04262 [math.CO]

  61. C. Haase, G. Musiker, J. Yu, Linear systems on tropical curves. Math. Z. 270(3-4), 1111–1140 (2012) [ ​MR 2892941]

    Google Scholar 

  62. P.L. Hammer, A.K. Kelmans, Laplacian spectra and spanning trees of threshold graphs. Discret. Appl. Math. 65(1-3), 255–273 (1996). First International Colloquium on Graphs and Optimization (GOI), 1992 (Grimentz) [ ​MR 1380078 (97d:05205)]

    Google Scholar 

  63. A. Hatcher, Algebraic Topology (Cambridge University Press, Cambridge, 2002) [ ​MR 1867354 (2002k:55001)]

    MATH  Google Scholar 

  64. J. Herzog, E.M. Li Marzi, Bounds for the Betti numbers of shellable simplicial complexes and polytopes, in Commutative Algebra and Algebraic Geometry (Ferrara). Lecture Notes in Pure and Application Mathematic, vol. 206 (Dekker, New York, 1999), pp. 157–167 [ ​MR 1702104 (2001b:13017)]

    Google Scholar 

  65. J. Herzog, A survey on Stanley depth, in Monomial Ideals, Computations and Applications, Lecture Notes in Mathematics, vol. 2083 (Springer, Heidelberg, 2013), pp. 3–45 [ ​MR 3184118]

    Google Scholar 

  66. J. Hladký, D. Krá ’l, S. Norine, Rank of divisors on tropical curves. J. Comb. Theory Ser. A 120(7), 1521–1538 (2013) [ ​MR 3092681]

    Google Scholar 

  67. A. Joyal, Une théorie combinatoire des séries formelles. Adv. Math. 42(1), 1–82 (1981) [ ​MR 633783]

    Google Scholar 

  68. G. Kalai, Enumeration of Q-acyclic simplicial complexes. Israel J. Math. 45(4), 337–351 (1983) [ ​MR 720308 (85a:55006)]

    Google Scholar 

  69. L. Katthän, Stanley depth and simplicial spanning trees. J. Algebraic Combin. 42(2), 507–536 (2015) [ ​MR 3369566]

    Google Scholar 

  70. G. Kirchhoff, Über die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Verteilung galvanischer Ströme geführt wird. Ann. Phys. Chem. 72 497–508 (1847)

    Article  Google Scholar 

  71. V. Klee, C. Witzgall, Facets and vertices of transportation polytopes, Mathematics of the Decision Sciences, Part I (Seminar, Stanford, Calif., 1967), (American Mathematical Society, Providence, RI, 1968), pp. 257–282 [ ​MR 0235832]

    Google Scholar 

  72. W. Kook, V. Reiner, D. Stanton, Combinatorial Laplacians of matroid complexes. J. Am. Math. Soc. 13(1), 129–148 (2000) [ ​MR 1697094 (2001e:05028)]

    Google Scholar 

  73. W. Kook, K.-Ju Lee, A formula for simplicial tree numbers of matroid complexes, Europ. J. Combin., to appear (2015)

    Google Scholar 

  74. W. Kook, Weighted Tree-Numbers of Matroid Complexes, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015), Discrete Mathematics & Theoretical Computer Science Proceedings, AS, Association of Discrete Mathematics & Theoretical Computer Science, Nancy, pp. 709–720 (to appear, 2015)

    Google Scholar 

  75. V. Krushkal, D. Renardy, A polynomial invariant and duality for triangulations. Electron. J. Combin. 21(3), Paper 51, 16 pp. (2012) [ ​MR 3262279]

    Google Scholar 

  76. L. Levine, J. Propp, What is \(\ldots\) a sandpile?. Not. Am. Math. Soc. 57(8), 976–979 (2010) [ ​MR 2667495]

    Google Scholar 

  77. D.J. Lorenzini, A finite group attached to the Laplacian of a graph. Discret. Math. 91(3), 277–282 (1991) [ ​MR 1129991 (93a:05091)]

    Google Scholar 

  78. Y. Luo, Rank-determining sets of metric graphs. J. Comb. Theory Ser. A 118(6), 1775–1793 (2011) [ ​MR 2793609 (2012d:05122)]

    Google Scholar 

  79. R. Lyons, Random complexes and \(l^{2}\)-Betti numbers. J. Topol. Anal. 1(2), 153–175 (2009) [ ​MR 2541759 (2010k:05130)]

    Google Scholar 

  80. N.V.R. Mahadev, U.N. Peled, Threshold Graphs and Related Topics, Annals of Discrete Mathematics, vol. 56 (North-Holland Publishing Co., Amsterdam, 1995) [ ​MR 1417258 (97h:05001)]

    MATH  Google Scholar 

  81. J.L. Martin, V. Reiner, Factorization of some weighted spanning tree enumerators. J. Comb. Theory Ser. A 104(2), 287–300 (2003) [ ​MR 2019276 (2004i:05070)]

    Google Scholar 

  82. J.L. Martin, V. Reiner, Cyclotomic and simplicial matroids. Israel J. Math. 150, 229–240 (2005) [ ​MR 2255809 (2007g:05040)]

    Google Scholar 

  83. J.L. Martin, M. Maxwell, V. Reiner, S.O. Wilson, Pseudodeterminants and perfect square spanning tree counts. J. Comb. 6(3), 295–325 (2015) [ ​MR 3357126]

    Google Scholar 

  84. G. Masbaum, A. Vaintrob, A new matrix-tree theorem. Int. Math. Res. Not. 27, 1397–1426 (2002) [ ​MR 1908476 (2003a:05107)]

    Google Scholar 

  85. M. Maxwell, Enumerating bases of self-dual matroids. J. Comb. Theory Ser. A 116(2), 351–378 (2009) [ ​MR 2475022 (2010a:05048)]

    Google Scholar 

  86. D. Mayhew, Equitable matroids. Electron. J. Comb. 13(1), Research Paper 41, 8 pp. (electronic) (2006) [ ​MR 2223516 (2007c:05047)]

    Google Scholar 

  87. C. Merino, The chip-firing game. Discret. Math. 302(1-3), 188–210 (2005) [ ​MR 2179643 (2007c:91036)]

    Google Scholar 

  88. R. Merris, Degree maximal graphs are Laplacian integral. Linear Algebra Appl. 199, 381–389 (1994) [ ​MR 1274427 (95e:05083)]

    Google Scholar 

  89. J. Milnor, Whitehead torsion. Bull. Am. Math. Soc. 72, 358–426 (1966) [ ​MR 0196736 (33 #4922)]

    Google Scholar 

  90. J.W. Moon, Counting Labelled Trees, From lectures delivered to the Twelfth Biennial Seminar of the Canadian Mathematical Congress (Vancouver, vol. 1969, (Canadian Mathematical Congress, Montreal, Quebec, 1970) [ ​MR 0274333 (43 #98)]

    Google Scholar 

  91. G. Musiker, V. Reiner, The cyclotomic polynomial topologically. J. Reine Angew Math. 687, 113–132 (2014) [ ​MR 3176609]

    Google Scholar 

  92. J. Oxley, Matroid Theory, Oxford Graduate Texts in Mathematics, 2nd ed. vol. 21 (Oxford University Press, Oxford, 2011) [ ​MR 2849819 (2012k:05002)]

    Google Scholar 

  93. D. Perkinson, J. Perlman, J. Wilmes, Primer for the Algebraic Geometry of Sandpiles, Tropical and non-Archimedean geometry, Contemporary Mathematics, vol. 605 (American Mathematical Society, Providence, RI, 2013), pp. 211–256 [ ​MR 3204273]

    Google Scholar 

  94. A. Petersson, Enumeration of spanning trees in simplicial complexes. Master’s thesis, Uppsala Universitet (2009)

    Google Scholar 

  95. M.R. Pournaki, S.A. Seyed Fakhari, M. Tousi, S. Yassemi, What is \(\ldots\) Stanley depth?. Not. Am. Math. Soc. 56(9), 1106–1108 (2009) [ ​MR 2568497 (2010k:05346)]

    Google Scholar 

  96. H. Prüfer, Neuer beweis eines satzes über permutationen. Arch. Math. Phys. 27, 142–144 (1918)

    MATH  Google Scholar 

  97. G.A. Reisner, Cohen-Macaulay quotients of polynomial rings. Adv. Math. 21(1), 30–49 (1976) [ ​MR 0407036 (53 #10819)]

    Google Scholar 

  98. J.B. Remmel, S. Gill Williamson, Spanning trees and function classes. Electron. J. Comb. 9(1), Research Paper 34, 24 pp. (electronic) (2002) [ ​MR 1928786 (2003g:05067)]

    Google Scholar 

  99. H.I. Scoins, The number of trees with nodes of alternate parity. Proc. Cambridge Philos. Soc. 58, 12–16 (1962) [ ​MR 0136554 (25 #24)]

    Google Scholar 

  100. M.A. Simmonard, G.F. Hadley, The maximum number of iterations in the transportation problem. Naval Res. Logistics Q. 6, 125–129 (1959) [ ​MR 0108401 (21 #7117)]

    Google Scholar 

  101. R.P. Stanley, Linear Diophantine equations and local cohomology. Invent. Math. 68(2), 175–193 (1982) [ ​MR 666158 (83m:10017)]

    Google Scholar 

  102. R.P. Stanley, A combinatorial decomposition of acyclic simplicial complexes. Discret. Math. 120(1-3), 175–182 (1993) [ ​MR 1235904 (94k:55027)]

    Google Scholar 

  103. R.P. Stanley, Combinatorics and Commutative Algebra, 2nd ed., Progress in Mathematics, vol. 41 (Birkhäuser Boston, Inc., Boston, MA, 1996) [ ​MR 1453579 (98h:05001)]

    Google Scholar 

  104. R.P. Stanley, Enumerative Combinatorics, Vol. 2. Cambridge Studies in Advanced Mathematics, vol. 62 (Cambridge University Press, Cambridge, 1999). With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin. [ ​MR 1676282 (2000k:05026)]

    Google Scholar 

  105. R.P. Stanley, An Introduction to Hyperplane Arrangements, Geometric combinatorics, IAS/Park City Math. Ser., vol. 13, (American Mathematical Society, Providence, RI, 2007), pp. 389–496 [ ​MR 2383131]

    Google Scholar 

  106. Y. Su, D.G. Wagner, The lattice of integer flows of a regular matroid. J. Comb. Theory Ser. B 100(6), 691–703 (2010) [ ​MR 2718687 (2012a:05066)]

    Google Scholar 

  107. J.J. Sylvester, On the change of systems of independent variables. Quart. J. Math. 1 42–56 (1857). [Collected Mathematical Papers, vol. 2, Cambridge, 1908, pp. 65–85]

    Google Scholar 

  108. W. T. Tutte, On the Spanning Trees of Self-Dual Maps, Second International Conference on Combinatorial Mathematics (New York, 1978), Annals of the New York Academy of Sciences, vol. 319, (New York Academy of Sciences, New York, 1979), pp. 540–548 [ ​MR 556066 (81c:05031)]

    Google Scholar 

  109. D.B. Wilson, Generating Random Spanning Trees More Quickly than the Cover Time, in Proceedings of the Twenty-eighth Annual ACM Symposium on the Theory of Computing (Philadelphia, PA, 1996), (ACM, New York, 1996), pp. 296–303 [ ​MR 1427525]

    Google Scholar 

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Acknowledgements

We thank our colleagues Amy Wagler and Jennifer Wagner for valuable suggestions regarding exposition and clarity.

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Texas at El Paso, El Paso, TX, USA

    Art M. Duval

  2. Division of Applied Mathematics and Department of Computer Science, Brown University, Providence, RI, USA

    Caroline J. Klivans

  3. Department of Mathematics, University of Kansas, Lawrence, KS, USA

    Jeremy L. Martin

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Editors and Affiliations

  1. Dept of Math, Statistics & Comp.Sc, Macalester College, St. Paul, Minnesota, USA

    Andrew Beveridge

  2. Dept. of Mathematics, University of South Carolina, Columbia, South Carolina, USA

    Jerrold R. Griggs

  3. Dept. of Mathematics, Iowa State University, Ames, Iowa, USA

    Leslie Hogben

  4. School of Mathematics, University of Minnesota, Minneapolis, Minnesota, USA

    Gregg Musiker

  5. School of Mathematics & Computer Sc, Georgia Institute of Technology, Atlanta, Georgia, USA

    Prasad Tetali

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Duval, A.M., Klivans, C.J., Martin, J.L. (2016). Simplicial and cellular trees. In: Beveridge, A., Griggs, J., Hogben, L., Musiker, G., Tetali, P. (eds) Recent Trends in Combinatorics. The IMA Volumes in Mathematics and its Applications, vol 159. Springer, Cham. https://doi.org/10.1007/978-3-319-24298-9_28

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