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ОбложкаPetersen T.K. Eulerian numbers. - New York: Birkhauser, 2015. - xviii, 456 p.: ill., tab. - (Birkhauser Advanced Texts Basler Lehrbucher). - Bibliogr.: p.347-358. - Ind.: p.453-456. - ISBN 978-1-4939-3090-6; ISSN 1019-6242
Шифр: (И/В17-Р50) 02
 

Место хранения: 02 | Отделение ГПНТБ СО РАН | Новосибирск

Оглавление / Contents
 
Part I Combinatorics

1  Eulerian numbers ............................................. 3
   1.1  Binomial coefficients ................................... 3
   1.2  Generating functions .................................... 5
   1.3  Classical Eulerian numbers .............................. 6
   1.4  Eulerian polynomials .................................... 9
   1.5  Two important identities ............................... 10
   1.6  Exponential generating function ........................ 12
   Problems .................................................... 14
2  Narayana numbers ............................................ 19
   2.1  Catalan numbers ........................................ 19
   2.2  Pattern-avoiding permutations .......................... 20
   2.3  Narayana numbers ....................................... 23
   2.4  Dyck paths ............................................. 26
        2.4.1  Counting all Dyck paths ......................... 27
        2.4.2  Counting Dyck paths by peaks .................... 29
        2.4.3  A bijection with 231-avoiding permutations ...... 31
   2.5  Planar binary trees .................................... 34
   2.6  Noncrossing partitions ................................. 36
   Problems .................................................... 40
3  Partially ordered sets ...................................... 47
   3.1  Basic definitions and terminology ...................... 47
   3.2  Labeled posets and P-partitions ........................ 50
   3.3  The shard intersection order ........................... 54
   3.4  The lattice of noncrossing partitions .................. 57
   3.5  Absolute order and Noncrossing partitions .............. 61
   Problems .................................................... 64
4  Gamma-nonnegativity ......................................... 71
   4.1  The idea of gamma-nonnegativity ........................ 71
   4.2  Gamma-nonnegativity for Eulerian numbers ............... 72
   4.3  Gamma-nonnegativity for Narayana numbers ............... 76
   4.4  Palindromicity, unimodality, and the gamma basis ....... 77
   4.5  Computing the gamma vector ............................. 80
   4.6  Real roots and log-concavity ........................... 81
   4.7  Symmetric boolean decomposition ........................ 84
   Problems .................................................... 88
5  Weak order, hyperplane arrangements, and the Tamari
   lattice ..................................................... 95
   5.1  Inversions ............................................. 95
   5.2  The weak order ......................................... 98
   5.3  The braid arrangement ................................. 100
   5.4  Euclidean hyperplane arrangements ..................... 102
   5.5  Products of faces and the weak order on chambers ...... 105
   5.6  Set compositions ...................................... 108
   5.7  The Tamari lattice .................................... 113
   5.8  Rooted planar trees and faces of the associahedron .... 115
   Problems ................................................... 123
6  Refined enumeration ........................................ 127
   6.1  The idea of a q-analogue .............................. 127
   6.2  Lattice paths by area ................................. 129
   6.3  Lattice paths by major index .......................... 132
   6.4  Euler-Mahonian distributions .......................... 134
   6.5  Descents and major index .............................. 137
   6.6  q-Catalan numbers ..................................... 139
   6.7  q-Narayana numbers .................................... 140
   6.8  Dyck paths by area .................................... 143
   Problems ................................................... 149
7  Cubes, Carries, and an Amazing Matrix: (Supplemental) ...... 151
   7.1  Slicing a cube ........................................ 151
   7.2  Carries in addition ................................... 154
   7.3  The amazing matrix .................................... 156

Part II  Combinatorial topology

8  Simplicial complexes ....................................... 163
   8.1  Abstract simplicial complexes ......................... 163
   8.2  Simple convex polytopes ............................... 166
   8.3  Boolean complexes ..................................... 167
   8.4  The order complex of a poset .......................... 169
   8.5  Flag simplicial complexes ............................. 170
   8.6  Balanced simplicial complexes ......................... 172
   8.7  Face enumeration ...................................... 173
   8.8  The h-vector .......................................... 175
   8.9  The Dehn-Sommerville relations ........................ 177
   Problems ................................................... 181
9  Barycentric subdivision .................................... 185
   9.1  Barycentric subdivision of a finite cell complex ...... 185
   9.2  The barycentric subdivision of a simplex .............. 187
   9.3  Brenti and Welker's transformation .................... 190
   9.4  The h-vector of sd(Δ) and j-Eulerian numbers ......... 193
   9.5  Gamma-nonnegativity of h(sd(Δ)) ....................... 196
   9.6  Real roots for barycentric subdivisions ............... 200
   Problems ................................................... 201
10 Characterizing ƒ-vectors: (Supplemental) ................... 203
   10.1 Compressed simplicial complexes ....................... 203
   10.2 Proof of the compression lemma ........................ 207
   10.3 Kruskal-Katona-Schiitzenberger inequalities ........... 215
   10.4 Frankl-Fiiredi-Kalai inequalities ..................... 219
   10.5 Multicomplexes and M-vectors .......................... 223
   10.6 The Stanley-Reisner ring .............................. 225
   10.7 The upper bound theorem and the g-theorem ............. 228
   10.8 Conjectures for flag spheres 230

Part III Coxeter groups

11 Coxeter groups ............................................. 237
   11.1 The symmetric group ................................... 237
   11.2 Finite Coxeter groups: generators and relations ....... 243
   11.3 IV-Mahonian distribution .............................. 246
   11.4 VK-Eulerian numbers ................................... 246
   11.5 Finite reflection groups and root systems ............. 250
        11.5.1 Type An-1 ...................................... 254
        11.5.2 Type Bn ........................................ 254
        11.5.3 Type Cn ........................................ 255
        11.5.4 Type Dn ........................................ 255
        11.5.5 Roots for I2(m) ................................ 256
   11.6 The Coxeter arrangement and the Coxeter complex ....... 257
   11.7 Action of W and cosets of parabolic subgroups ......... 259
   11.8 Counting faces in the Coxeter complex ................. 262
   11.9 The W-Euler-Mahonian distribution ..................... 264
   11.10 The weak order ....................................... 266
   11.11 The shard intersection order ......................... 269
   Problems ................................................... 271
12 W-Narayana numbers ......................................... 273
   12.1 Reflection length and Coxeter elements ................ 273
   12.2 Absolute order and W-noncrossing partitions ........... 276
   12.3 W-Catalan and W-Narayana numbers ...................... 277
   12.4 Coxeter-sortable elements ............................. 280
   12.5 Root posets and W-nonnesting partitions ............... 282
   12.6 The W-associahedron ................................... 287
   Problems ................................................... 290
13 Combinatorics for Coxeter groups of types Bn and Dn:
   (Supplemental) ............................................. 293
   13.1 Type Bn Eulerian numbers .............................. 293
   13.2 Type Bn gamma-nonnegativity ........................... 297
   13.3 Type Dn Eulerian numbers .............................. 301
   13.4 Type Dn gamma-nonnegativity ........................... 303
   13.5 Combinatorial models for shard intersections .......... 307
        13.5.1 Type An-1 ...................................... 307
        13.5.2 Type Bn ........................................ 311
        13.5.3 Type Dn ........................................ 316
   13.6 Type Bn noncrossing partitions and Narayana numbers ... 320
   13.7 Gamma-nonnegativity for Cat(Bn;t) ..................... 325
   13.8 Type Dn noncrossing partitions and Narayana numbers ... 327
   13.9 Gamma-nonnegativity for Cat(Dn; t) .................... 330
14 Afflne descents and the Steinberg torus: (Supplemental) .... 333
   14.1 Affine Weyl groups .................................... 333
   14.2 Faces of the affine Coxeter complex ................... 334
   14.3 The Steinberg torus ................................... 338
   14.4 Affine Eulerian numbers ............................... 341
        14.4.1 Type An-1 ...................................... 341
        14.4.2 Type Bn ........................................ 342
        14.4.3 Type Cn ........................................ 343
        14.4.4 Type Dn ........................................ 344
References .................................................... 347
Hints and Solutions ........................................... 359
Index ......................................................... 453

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