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Index

, see Chen, William Yongchuan Ardila, Federico, 168 , see Deng, Eva Yuping Armstrong, Drew, 147, 148 , see Du, Rosena Ruoxia arrangement, Catalan, see Catalan , see Hou, Sharon Jiangxia arrangement , see Hu, Yangzhou ascent sequence, restricted, 73 , see Jing An Asinowski, Andrei, 60 , see Luo, Jianjin associahedron, 159 , see Ming’antu asymptotic density, 135 , see Wei, Fan upper, 135 , see Xin, Guoce Athanasiadis, Christos A., 89, 95 , see Yan, Huafei Aval, Jean-Christophe, 96, 142 , see Zhang, Yan (a,b)-Catalan number, see Catalan number, Bacher, Roland, 160 (a,b) Baek, Jineon, 152 Abel’s theorem, see theorem, Abel’s Baldoni-Silva, Maria Welleda, 97, 150 Abramson, Morton, 166 ballot number, see number, ballot Acerbi, Fabio, 127 ballot problem, see problem, ballot Ackerman, Eyal, 166 ballot sequence, see sequence, ballot Aebi, Christian, 188 Banderier, Cyril, 61 Aguiar, Marcelo, 72, 100 Barbier, Joseph Emile,´ 184 Aigner, Martin, 162, 187 Barcucci, Elena, 63 Albert, Michael Henry, 73 Barequet, Gill, 166 algebra, Temperley-Lieb, 142 Baxter permutation, see permutation, Baxter algebraic language, see language, algebraic Becker, Harold W., 87, 168 alternating Baxter permutation, Beckwith, David, 89, 160 see permutation, Baxter, alternating Bell, Eric Temple, 186 alternating tree, see tree, alternating Berenstein, Arkady, 143 Ampere,` Andre-Marie,´ 185 Bergeron, Franc¸ois, 142 Anderson, Jaclyn Ann, 77 Bergeron, Nantel, 142 Andre,´ Desir´ e,´ 138, 184 Berkolaiko, Gregory, 151 Andrews, George W. Eyre, 92, 153, 160 Berlekamp, Elwyn Ralph, 153 animal, directed, 167 Bernardi, Olivier, 146 arcs Bernhardt, Frank, 56 noncrossing, 29 Bertram’s ballot theorem, 139 nonintersecting, 28 Bertrand, Joseph Louis Franc¸ois, 139, 184

205

206 Index

Bessenrodt, Christine, 96, 154, 160 infinitely many interpretations, 69 β(1,0)-tree, see tree, β(1,0) not always the solution, 152 Betr´ ema,´ Jean, 95 q-analogue, 121 Bevan, David, 70 (q,t), 83, 161 bijective proof, 7 super, 110 Billera, Louis Joseph, 87, 99 ternary, 148 Billey, Sara Cosette, 71, 76, 78, 80, 91, 142, Catalan simplicial set, 94 167 Catalan tree, see tree, Catalan binary tree, see tree, binary Catalan triangulation, see triangulation, Binet, Jacques Philippe Marie, 182 Catalan binomial coefficient, 3 Catalan, Eugene` Charles, 139, 148, 181 binomial theorem, generalized, 3 Catalania, 56 Bisch, Dietmar Herbert, 147 Cayley, Arthur, 160, 183 Bizley, Michael Terence Lewis, 148 Ceballos, Cesar, 159 Bjorner,¨ Anders, 158 cell, Schubert, 167 Bloom, David Michael, 139 centered tunnel, 23, 63 Bona,´ Miklos,´ 58, 79, 167 Chan, Clara Sophia, 141, 150 Bonin, Joseph Edmond, 165, 168 Chan-Robbins-Yuen polytope, see polytope, bounce, 123 Chan-Robbins-Yuen bounce path, see path, bounce Chandon, Jean-Louis, 156 Bousquet, Michel, 58 Chapman, Robin, 63, 71, 145 Bousquet-Melou,´ Mireille, 61, 144, 167 Chen, William Yongchuan, 64, 68, 170 bracketing, 7, 16 chess problem, 103 of a word, arbitrary, 127 chess tableau, see tableau, chess Brendsdal, Egil, 60 Choi, Suyoung, 148 Brenti, Francesco, 150 chords, nonintersecting, 28, 126 Brown, William G., 185 Chow, Timothy Yi-Chung, 90 Bru, Bernard, 185 Chung Graham, Fan-Rong King, 164 Bruckman, Paul S., 149 Chung, Kai Lai, 139 Bruhat order, see order, Bruhat Chung-Feller theorem, see theorem, de Bruijn, Nicolaas Govert, 9 Chung-Feller Brunvoll, Jon, 60 Cigler, Johann, 160 Buckley, Mitchell, 94 Ciucu, Mihai Adrian, 165 Claesson, Anders Karl, 59, 64, 73, 81, 93 Cairns, Grant, 188 code, see permutation, code of Callan, David, 58–62, 65, 66, 69, 70, 72, coins, ways to stack, 49 74–76, 82, 89, 96 complementary minors, see minors, Cameron, Peter Jephson, 187 complementary Can, Mahir, 96 compositional inverse, 116 Carlitz, Leonard, 153, 160 Comtet, Louis, 168 Catalan alternative tableau, see tableau, cone, Littlewood-Richardson, 93 Catalan alternative conjugacy class, 104 Catalan arrangement, 95 conjugate (of a finite sequence), 12 Catalan disease, see disease, Catalan context-free language, see language, Catalan matroid, see matroid, Catalan context-free Catalan number continued fraction, 111, 122, 137 (a,b), 109 convex (n + 2)-gon, 1 algebraic interpretations, 103, 105 convex position, probability of, 118 asymptotic series, 169 convexotope, 118 definition, 1 Conway, John Horton, 96 fundamental recurrence, 3 core, 36

Index 207

Cori, Robert, 79, 80, 146 double rise (of a Schroder¨ path), see path, Coxeter, Harold Scott MacDonald, 96 Schroder,¨ double rise cyclic plane forest, see forest, cyclic plane doubly stochastic matrix, see matrix, doubly cyclic shift, 12 stochastic Cyvin, Bjørg N., 60 down step, see path, Dyck, down step Cyvin, Sven Josef, 60, 98 Drake, Brian C., 148 Dress, Andreas, 146 Davenport-Schinzel sequence, see sequence, Du, Rosena Ruoxia, 64, 69 Davenport-Schinzel Dukes, William Marks B., 73 De Saintes-Catherine, Myriam, 68 Dulucq, Serge, 67, 79, 80, 146, 163, 167 Dean, Richard Albert, 92 Dumitriu, Ioana, 146 decomposable, see permutation, decomposable Dyck language, see language, Dyck deformation, versal, 94 Dyck path, see path, Dyck Del Lungo, Alberto, 63 Dyson, Freeman John, 169 Delest, Maylis, 67 Deng, Eva Yuping, 64, 69, 83 Edelman, Paul Henry, 87, 147, 158, 167 Denise, Alain, 61 edge, superfluous, 107 Deodhar, Vinay Vithal, 167 Egge, Eric Stephen, 83 depth first order, see order, depth first Eggleton, Roger Benjamin, 154 deque-sortable permutation, see permutation, Ehrhart polynomial, see polynomial, Ehrhart deque-sortable Elizalde, Sergi, 63, 70, 84, 85 Dershowitz, Nachum, 87 Eremenko, Alexandre, 151 Desarm´ enien,´ Jacques, 143 Eriksson, Henrik, 90 descent, 123 Errera, Andrea, 67 of a Dyck path, see path, Dyck, descent Etherington, Ivor Malcolm Haddon, 168 Deutsch, Emeric, 56, 57, 61–65, 70, 76, Eu, Sen-Peng, 151 83–86, 89, 93, 97, 99, 163, 168 Euler, Leonhard, 137, 178 Dewji, Rian, 93 Euler-Segner number, see number, diagram Euler-Segner Murasaki, 43 excedance Puttenham, 87 of a path, see path, excedance Young, 44 of a permutation, see permutation, paths inside, 116 excedance Dickins, Anthony Stewart Mackay, 141 excedance set, 76 difference operator, 112 dimer, 129 Fan, Chenteh Kenneth, 90 Dimitrov, Ivan, 93 Feller, William, 138, 139 direct sum decomposition, 124 Ferrari, Luca, 92, 94 directed animal, see animal, directed Fibonacci tree, see tree, Fibonacci Disanto, Filippo, 92, 94 Fishburn, Peter C., 92 disease, Catalan, 56, 188 flag, 106 dissection, of a convex polygon, 108, 119, flag variety, see variety, flag 121, 128 Flajolet, Philippe, 61, 148 Djokov´ıc, Dragomir Z, 143 flow polytope, see polytope, flow Dokos, Theodore, 80 f -matching, see matching, f - dominance order, see order, dominance Fomin, Sergey Vladimirovich, 80, 95, 96, 143 domino, 129 Fonseca, Tiago, 143 Donaghey, Robert James, 162 Forcey, Stefan, 60 Donnelly, Robert Garvin, 92 forest, cyclic plane, 170 Dosliˇ c,´ Tomislav, 98 Françon, Jean, 81 Doubilet, Peter Michael, 143 Frankel, Norman Edward, 169

208 Index

Freund, John E., 92 Gudmundsson, Hilmar Haukur, 73 Friedmann, Tamar, 143 Guibert, Olivier, 80, 163 frieze pattern, 96 guillotine rectangulation, see rectangulation, function guillotine Mobius,¨ 110, 155 Gutman, Ivan, 98 quasisymmetric, 104 Guy, Richard Kenneth, 66, 95, 154 Furlinger,¨ Johannes, 160 von Fuss, Paul Heinrich, 178 Hadamard product, see product, Hadamard Fuss, Nicolaus Ivanovich, 146, 178, 180 Haglund, James, 161 Fuss-Catalan number, see number, Haiman, Mark David, 83, 142, 159, 167 Fuss-Catalan half edge, 126 Hall, Jr., Marshall, 186 Gabrielov, Andrei, 151 Hammersley, John Michael, 78 Gao, Zhenguang, 169 de la Harpe, Pierre, 78, 142 Gardner, Martin, 88, 187 Harrison, Jonathan, 151 Gardy, Daniele,` 61 Hausdorff, Felix, 168 Garner, Richard Henry George, 94 Hetyei, Gabor, 99 Garrabrant, Scott, 152 Heubach, Silvia Petra, 147 Garsia, Adriano Mario, 83, 142 hexaﬂexagon, 82 Gelfand, Israil Moiseivich, 67, 157, 159 hexagonal lattice, see lattice, hexagonal generating tree, see tree, generating hill, see path, Dyck, hill genus 0, see permutation, genus 0 hill-free, see path, Dyck, hill-free Gessel, Ira Martin, 58, 80, 89, 95, 138, 148, Hipparchus, 127 162 Hivert, Florent, 73 Getu, Seyoum, 164 Hofbauer, Josef, 160 Gewurz, Daniele A., 81 Hoggatt, Jr., Verner Emil, 149, 164 Ghys, Etienne,´ 166 Holm, Thorsten, 96 Gili, Joan L., 141 Hopf algebra, 96 Gire, Sophie, 164, 167 Hopkins, Brian Preston, 87 Glasser, M. Lawrence, 169 horizontally convex, see polyomino, Gobet, Thomas, 74 horizontally convex Goldbach, Christian, 178 Hou, Sharon Jiangxia, 148 Goldberg, Lisa Robin, 141 Hu, Yangzhou, 73 Goodman, Frederick Michael, 78, 142 Huang, Samuel, 158 Gould, Henry Wadsworth, 88, 185, 186 Humphreys, Katherine, 184 Goulden, Ian Peter, 145, 167 Huq, Aminul, 139 Gouyou-Beauchamps, Dominique, 61, 163, h-vector, 121 167 hyperplane, 50 Graev, Mark Iosifovich, 67, 157 Graham, Ronald Lewis, 82, 164 ideal, of algebra of upper triangular matrices, graph 103 incomparability, 94 incomparability graph, see graph, noncrossing, 129 incomparability unit interval, 94 indecomposable, see permutation, Grassmannian, degree, 104 indecomposable Grossman, Howard D., 148 index, major, 122 group internal triangle, see triangle, internal Coxeter, 138 internal vertex, 16 symmetric, 104 intersection homology, 104 symplectic, 71, 103 invariant, of SL(2,C), 103 Grunert, Johann August, 182 inverse, of a bijection, 7

Index 209

inversion, Mobius,¨ 155 Kummer, Ernst Eduard, 168 involution Kung, Joseph PeeSin, 143 2143-avoiding, 127 Kuznetsov, Alexander, 162 ﬁxed-point free, 38 vexillary, 127 Labelle, Gilbert, 58 Lack, Stephen Geoffrey, 94 Jackson, David M. R., 79, 145 Lagrange inversion formula, 153, 182, 183 j-ary tree, see tree, j-ary Lakshmibai, Venkatramani, 167 Jiang, Maosen, 162 Lame,´ Gabriel Leon´ Jean Baptiste, 181 Jing An, 177 language Jockusch, William, 71, 76, 78, 91, 142 algebraic, 99 Johansson, Robert, 99, 166 context-free, 99 Jones, Michael Alan, 87 Dyck, 99 Jones, Vaughan Frederick Randal, 78, 142, Larcombe, Peter J., 178 147 Lascoux, Alain, 80 Jonsson, Jakob, 146 lattice Jørgensen, Peter, 96 hexagonal, 19 simplicial Eulerian, 119 Kafashan, Mohammadmehdi, 60 Tamari, 119 Kapranov, Mikhail M., 159 Young’s, 89 Kasraoui, Anisse, 68 lattice path, see path, lattice Kauers, Manuel, 169, 187 Lee, Carl William, 159, 160 k-divisible noncrossing partition, see partition, Lee, Kyungyong, 96 k-divisible, noncrossing left child, in a binary tree, 6 Keller, Gordon Ernest, 92 left subtree, in a binary tree, 5 Kellogg, Alta, 154 Lehmer, Derrick Henry, 169 Kepler tower, 31 Kim, Jang Soo, 60 Leighton, Frank Thomson, 98 king, on a chessboard, 131 Lemaire, Jacques, 156 King, Ronald Curtis, 71 Leroux, Pierre, 58 Kirkman, Thomas Penyngton, 160, 182 Levine, Jack, 67 Kirkman-Cayley number, see number, Levine, Lionel, 87 Kirkman-Cayley Lewis, Joel Brewster, 86 Kitaev, Sergey Vladimirovich, 59, 73, 81 Li, Li, 96 Klarner, David Anthony, 9, 56 Li, Nelson Y. , 147 Klazar, Martin, 72, 88, 163, 165, 167 Lieb, Elliot Hershel, 67 Kleiman, Mark, 164 linear extension, 46 Knuth, Donald Ervin, 9, 71, 77–80, 92, 99, 165 Linusson, Svante, 64, 80, 93, 99, 166 Koganov, Leonid M., 146 Liouville, Joseph, 181 Konvalinka, Matjaz,˘ 149 Liskovets, Valery Anisimovich, 145, 146, 168 Kooken, Jack H., 146 Littlewood-Richardson cone, see cone, Korner,¨ Janos,´ 152 Littlewood-Richardson Koshy, Thomas, 169, 188 Liu, Shu-Chung, 151 Kostant partition function, 97 Lloyd, E. Keith, 60 Kotelnikow, Semen¨ Kirillovich, 180 Loday, Jean-Louis, 96, 159 Krattenthaler, Christian Friedrich, 66, 68, 169 Loehr, Nicholas Anthony, 83, 96 Kremer, Darla, 164 loop percolation, 143 Kreweras, Germain, 162 Louchard, Guy, 67 k-triangulation, 107 Lucas, Franc¸ois Edouard´ Anatole, 185 Kube, Nate, 69 van de Lune, Jan, 154 Kummer’s theorem, see theorem, Kummer’s Luo, Jianjin, 178

210 Index

Macdonald, Ian Grant, 80 Mullin, Ronald Cleveland, 88, 146, 165 MacMahon, Percy Alexander, 78, 160, 162, Murasaki diagram, see diagram, Murasaki 185 Murasaki, Lady, 88 major index, see index, major Myangat, Sharabiin, 177 Maleki, Mehdi, 60 Malvenuto, Claudia, 152 Nagy, Gabor´ V., 153 mania, Catalan, 56 Nakamigawa, Tomoki, 146 Mansour, Touﬁk, 60, 86, 147 Narayana number, see number, Narayana map Narayana, Tadepalli Venkata, 138, 139, 162 rational, 104 natural correspondence, 9 rooted planar, 106 near semiring, see semiring, near matching necklace, 43 f -, 69 net (of chords), 114 noncrossing, 28 Netto, Eugen Otto Erwin, 185 nonnesting, 29 Newman, Morris, 98 matrix Newton, Isaac, 3 alternating sign Ngram chart, 187 132-avoiding, 131 Nica, Alexandru Mihai, 87 doubly stochastic, 113 Nichols, Warren Douglas, 139 positive deﬁnite, 53 Niederhausen, Heinrich, 70 tridiagonal, 53, 65 noncrossing graph, see graph, noncrossing matroid, Catalan, 134 noncrossing matching, see matching, McCabe, Adam R., 93 noncrossing McCammond, Jonathan Paul, 188 noncrossing partition, see partition, Mead, David Godfrey, 143 noncrossing de Medicis,´ Anne, 68 noncrossing/nonnesting analogy, 60 Merola, Francesca, 81 nonnesting matching, see matching, Mesz´ aros,´ Karola, 87, 149, 157, 158, 165 nonnesting method of images, 184 nonnesting partition, see partition, nonnesting de Mier, Anna, 168 Novaes, Marcel, 151 Miner, Samuel, 84 Noy, Marc, 168 Ming’antu, 177 N-rational, 115 Minggatu, 177 number minor, nonvanishing, 104 ballot, 102, 183 minors, complementary, 105 Catalan, see Catalan number Mirmanoff, Dmitry Semionovitch, 184 Euler-Segner, 186 Mobius¨ function, see function, Mobius¨ Fuss-Catalan, 108, 180, 183 Mobius¨ inversion, see inversion, Mobius¨ Kirkman-Cayley, 183 Mobius¨ band, 131 Motzkin, 125, 132, 163 Mohanty, Sri Gopal, 138, 139 Narayana, 124, 158 Monsky, Paul, 143 ordered Bell, 183 Moon, John W., 93 Pfaff-Fuss-Catalan, 146 Moreira, Walter, 72 Schroder,¨ 110, 127, 131 Morselt, B. J. M., 9 q-analogue, 165 Moser, William Oscar Jules, 166 Segner, 186 Moszkowski, Paul, 162 winding, 106 Motzkin number, see number, Motzkin Motzkin path, see path, Motzkin Oakley, Cletus Odia, 82 Motzkin, Theodore Samuel, 162 Odlyzko, Andrew Michael, 95 Moulton, Vincent, 146 Olsson, Jørn Børling, 78 Muller,¨ Thomas W., 169 operation

Index 211

commutative nonassociative binary, 133 double rise, 26 nonassociative binary, 8 small, 164 order valley, 25 Bruhat, 132 Paule, Peter, 187 depth first, 10 peak (of a Dyck path), see path, Dyck, peak dominance, 36 Peart, Paul, 61 series-parallel interval, 92 Pechenik, Oliver, 166 unit interval, 92 Penaud, Jean-Guy, 67, 95 weak (Bruhat), 49 Pergola, Elisa, 63, 163, 164 ordered Bell number, see number, ordered Bell permutation ordered tree, see tree, ordered 132-avoiding, 80 orthonormal basis, 117 fixed point, 41 2-stack sortable, 131 Pak, Igor Markovich, 80, 84, 139, 152, 162 213-avoiding, 72 Pallo, Jean Marcel, 158 231-avoiding, 79 parallelogram polyomino, see polyomino, 312-avoiding, 37, 72, 79, 109 parallelogram 321-avoiding, 37, 71, 90, 142 Parcheesi, 87 alternating, 42, 152 parenthesization, 7, 16, 133 excedance, 41 Park, Hanchul, 148 fixed point, 41 partition 4231 and 3412-avoiding, 132 231-avoiding, 44 Baxter, 38, 90 noncrossing, 43, 44, 127, 129 alternating, 80 k-divisible, 109 code of, 71 strongly, 162 decomposable, 39 nonnesting, 44 deque-sortable, 165 for a reflection group, 89 dominant, 80 plane, see plane partition excedance, 41 pat, 40 genus 0, 37, 40, 41, 127 path indecomposable, 39 bounce, 123 separable, 166 Dyck, 20 smooth, 133 ascent, 62 sortable on two parallel queues, 38 definition, 7 stack-sortable, 37 descent, 22, 62 Pfaff, Johann Friedrich, 146 down step, 11 Pfaff-Fuss-Catalan number, see number, elevated, 90 Pfaff-Fuss-Catalan hill, 24 Pinter, Ron Yair, 166 hill-free, 62 Pinzani, Renzo, 63, 92, 94, 163 peak, 21 planar rhyme scheme, 87 rotated, 123 plane partition, 108 up step, 11 column-strict, 45 valley, 62 plane ternary tree, see tree, plane ternary excedance, 102 plane tree, see tree, plane lattice, 7 planted tree, see tree, planted in the plane, 20, 50, 126, 128 Plouffe, Simon, 188 nonintersecting, 146 Polya,´ George (Gyorgy),¨ 67 pair of, 27, 28 polynomial Motzkin, 24, 126 Ehrhart, 119 two-colored, 63, 96 q-Catalan, 124 Schroder,¨ 25, 164 wheel, 143

212 Index

polyomino Reading, Nathan Paul, 81, 143 enumeration, 67 rectangulation, 130 horizontally convex, 52 guillotine, 130 parallelogram, 66, 84, 128 reflection method, 184 symmetric, 31 reflection principle, 138 polytope Reifegerste, Astrid, 70, 76, 85, 97 Chan-Robbins-Yuen, 150 Reiner, Victor Schorr, 158, 167 flow, 97 relation, similarity, 93 secondary, 159 relative volume, see volume, relative Pomerance, Carl Bernard, 169 Remmel, Jeffrey Brian, 73 Poschel,¨ Reinhard, 168 Renault, Marc S., 184 poset representation, sign, 104 of direct sum decompositions, 124 restricted ascent sequence, see ascent of intervals of a chain, 46 sequence, restricted Postnikov, Alexander Yevgenievich, 67, 77, restricted growth function, 72 89, 95, 97, 99, 149, 150, 155, 157, 159, Retakh, Vladimir Solomonovich, 143 162 Reutenauer, Christophe, 80, 143, 160 Pouget, Jean, 156 Rhoades, Brendon Patrick, 143 Poupard, Yves, 162 Rice, John Michael, 141 preorder, 10 Richards, Dana Scott, 79 Preville-Ratelle,´ Louis-François, 158 right child, in a binary tree, 6 problem right subtree, in a binary tree, 6 ballot, 184 Rinaldi, Simone, 92, 94 Wedderburn-Etherington commutative Riordan, John F., 67, 185, 186 bracketing, 168 Robbins, David Peter, 150 Proctor, Robert Alan, 97 Robertson, Aaron, 84 Prodinger, Helmut, 87 Rodrigues, Benjamin Olinde, 181 product, Hadamard, 118 Rogers, Douglas G. E. D., 67, 93, 165 Propp, James Gary, 95 Romik, Dan, 143 Prouhet, Eugene,` 160 Ronco, Maria, 96 Pun, Siu Kei, 89 root system, 104 Puttenham diagram, see diagram, Puttenham root vertex Puttenham, George, 87 of a binary tree, 5 Puusa, Arto, 141 of a plane tree, 6 rooted planar map, see map, rooted planar q-analogue, 121 Rosas, Mercedes Helena, 97 q-binomial coefficient, 122 Roselle, David Paul, 88, 153 q-Catalan polynomial, see polynomial, Rota, Gian-Carlo, 143 q-Catalan rotated Dyck path, see path, Dyck, rotated (q,t)-Catalan number, see Catalan number, Rotem, Doron, 78 (q,t) Roth, Mike, 93 quantum Knizhnik-Zamolodchikov equation, Rowe, R. C., 183 143 RSK algorithm, 78 quasipolynomial, Ehrhart, 118 Ruskey, Frank, 69 quasisymmetric function, see function, Ruskuc,ˇ Nikola, 73 quasisymmetric Ryan, Kevin M., 167 Ryser, Herbert John, 185 Radoux, Christian, 154 Ramanujan, Srinivasa Aiyangar, 160 Sagan, Bruce Eli, 66, 149, 168 Rassart, Etienne,´ 146 Sam, Steven V., 93 rational map, see map, rational Sandhya, Baskhar, 167

Index 213

Santos Leal, Franciso, 159 Speicher, Roland, 87 Saracino, Daniel Harrison, 84 Stankova-Frenkel, Zvezdelina, 168 Schaeffer, Gilles, 144, 162 Stanley, Richard Peter, 64, 69, 71, 75, 77, 78, Schlosser, Martin, 68 80, 85, 88, 91, 95, 97, 99, 127, 141–143, Schmidt, Frank Whitney, 79 145, 149, 150, 152, 154, 157, 158, Schroder,¨ Friedrich Wilhelm Karl Ernst, 185 160–162 Schroder¨ number, see number, Schroder¨ Stanton, Dennis Warren, 78 Schroder¨ path, see path, Schroder¨ Stanton, Ralph Gordon, 88, 165 Schubert cell, see cell, Schubert Stein, Joel Alvin, 143 Schubert variety, see variety, Schubert Stein, Paul R., 186 Schur function, 141 Steinberg, Robert, 88 Schur ring, 133 Steingr´ımsson, Einar, 59 Schutzenberger,¨ Marcel-Paul, 80 Stembridge, John Reese, 14, 80, 93 Scoville, Richard A., 153 Stephens, Arthur Brooke, 163, 164, 166 secondary polytope, see polytope, secondary Stevens, Jan, 73, 94 von Segner, Johann Andreas, 178, 179 Stigler’s Law of Eponymy, 186 Segner number, see number, Segner Strayer, Michael, 60 semicircle law, 168 Street, Ross Howard, 94 semiorder, 92, 119 Strehl, Volker, 93 semiring, near, 55 strokes, minimum number, 106 separable permutation, see permutation, Stump, Christian, 73, 146 separable subtree (of a plane tree), see tree, plane, sequence subtree ballot, 6, 32 Sulanke, Robert A., 63, 67, 162, 164 strict, 12 Sundaram, Sheila, 71 convex, 118 Sunik, Zoran, 75 Davenport-Schinzel, 165 super Catalan number, see Catalan number, unimodal, 123 super series-parallel interval order, see order, superﬂuous edge, see edge, superﬂuous series-parallel interval symmetric group, see group, symmetric serieshelpmate, 103 symplectic group, see group symplectic Serrano, Luis Guillermo, 146 Seshadri, Conjeeveram Srirangachari, 167 table, inversion, 71 set, valid m-, 94 tableau Shapiro, Louis Welles, 56, 57, 59, 64, 67, 93, Catalan alternative, 96 98, 139, 141, 153, 162–166, 170 chess, 90 Shi, Jian-yi, 142 standard Young Shikibu, Murasaki, 88 of shape (n,n),44 sign representation, see representation, sign pair of, 45 sign type, ⊕, 104 Takacs,´ Lajos, 138, 184 similarity relation, see relation, similarity Tale of Genji, The,88 Simion, Rodica Eugenia, 76, 79, 87, 163, 165 Tamari lattice, see lattice, Tamari Simonyi, Gabor,´ 152 Tamari, Dov, 158 simplicial Eulerian lattice, see lattice, Tamm, Ulrich, 182 simplicial Eulerian Tarjan, Robert Endre, 80 Skandera, Mark Allan, 143 Taylor, Henry Martyn, 183 Sloane, Neil James Alexander, 187, 188 Temperley, Harold Neville Vazeille, 67 Smiley, Len, 58 Temperley-Lieb algebra, see algebra, Smith Barnes, Camillia, 82 Temperley-Lieb Smith normal form, 122 ternary Catalan number, see Catalan number, smooth permutation, see permutation, smooth ternary

214 Index

Terquem, Olry, 181 Vais¨ anen,¨ Kauko, 141 Tewari, Vasu, 148 valid m-set, see set, valid m- theorem valley (of a Dyck path), see path, Dyck, Abel’s, 136 valley Chung-Feller, 139 valley (of a Schroder¨ path), see path, Schroder,¨ Kummer’s, 134, 135 valley Zaslavsky’s, 96 Valtr, Pavel, 154, 156 Thomas, Hugh Ross, 98 van Lint, Jacobus Hendricus, 73 Tischler, David D., 142 variety, ﬂag, 167 Touchard, Jacques, 67 variety, Schubert, 104 tree Vatter, Vincent Russell, 73 alternating, 68, 157 Vauquelin, Bernard, 163 noncrossing, 68, 157 Vergne, Michele,` 97, 150 β(1,0),59 versal deformation, see deformation, versal binary, 5, 16, 127 Vetta, Adrian Roshan, 88 complete, 16 vexillary involution, see involution, vexillary Hopf algebra, 96 Viennot, Gerard´ Xavier, 67, 68, 71, 80, 81, 95, Catalan, 6 96, 146, 158, 167 Fibonacci, 109 volume, relative, 119 generating, 61, 164 increasing Wachs, Michelle Lynn, 158 noncrossing, 19 walk, on N, 126 nonnesting, 19 Walsh, Timothy Robert Stephen, 146 j-ary, 108 Wasanists, 88 ordered, 6 weak (Bruhat) order, see order, weak (Bruhat) plane, 6, 16, 126, 127, 183 Wedderburn, Joseph Henry Maclagan, 168 bicolored, 18 wedge, 56 subtree, 6 Wehlau, David Louis, 93 trivalent, 16 Wei, Fan, 166 plane ternary, 110 Welker, Volkmar, 161 planted, 16 West, Julian, 61, 79, 164–167 triangle, internal, 102 wheel polynomial, see polynomial, wheel triangulation Whitworth, William Allen, 139, 183 Catalan, 131 Wigner, Eugene Paul (JenoP´ al),´ 73, 75, 168 of a convex polygon, 1 Wilf, Herbert Saul, 95 of a polygon, 15, 96 Wilson, Jennifer, 93 of conv{0,ei − ej}, 157 winding number, see number, winding of the Mobius¨ band, 131 Wine,RussellL.,92 tridiagonal matrix, see matrix, tridiagonal Wisner, Robert Joel, 82 Trotter, Jr., William Thomas, 92 Woan, Wen-Jin, 61, 67 Tutte, William Thomas, 145 Woo, Alexander, 96, 142 two-colored Motzkin path, see path, Motzkin, Wormald, Nicholas Charles, 162 two-colored Xin, Guoce, 144 Ullman, Daniel Howard, 87 unimodal, 123 Yakoubov, Sophia, 147 unit interval graph, see graph, unit interval Yan, Catherine Huafei, 64, 69 unit interval order, see order, unit interval Yeh, Yeong-Nan, 151 up step, see path, Dyck, up step Young diagram, see diagram, Young upper asymptotic density, see asymptotic Young, Benjamin J., 99 density, upper Yuen, David ShunWing, 150

Index 215

Zaks, Shmuel, 87 Zeilberger, Doron, 84, 138, 143, 150, 166, 188 Zanello, Fabrizio, 78 Zelevinsky, Andrei Vladlenovich, 159 Zang, Nan, 82 Zeng, Jiang, 68 Zare, Douglas J., 168 Zhang, Yan, 157 Zaslavsky’s theorem, see theorem, Zaslavsky’s Ziegler, Gunter,¨ 159 zebra, 164 Zinn-Justin, Paul, 143