CoursE  X.V.  IMSc  2016

 
 

Course  IMSc  Chennai, India 

January-March 2016

An introduction to enumerative, algebraic and

bijective combinatorics


Index



admissible orientation (of a planar graph), Ch5b  148

algebra of formal power series  Ch1a  33

algebraic grammar, Ch1d  70

algebraic language, Ch1d  71

algebraic power series, Ch1b  27

(alpha)-distribution (in the Catalan garden), Ch2c  13, Ch2c  47, Ch2c  55-57

alternative tableaux, Ch4d-complements  6

alternating permutations, Ch3b  65

ambiguous grammar, Ch1d  74

André, Désiré, Ch3b  65

        André permutation, Ch4a  102

ansatz (cellular), Ch4d  50, Ch4d-complements  18

Appell polynomial, Ch3b  40

Arbogast triangle, Ch2c  17-19

arborescence (species), Ch3a  15, 44-45, 55

            (enriched increasing), Ch3b-complements  23

            (1-2 increasing), Ch4a  97

arctic circle theorem, Ch5b  130

area parameter (q-Catalan), Ch2d  59-62

Askey tableau (of orthogonal polynomials), Ch3b-complements  45

assemblée (of structures for species), Ch3a  28

            (of permutations), Ch3b  5

            (of increasing arborescences), Ch4a  95

associahedron, Ch4d-complements  44, Ch5b  116

automaton, Ch1d  77

average cost (of an insertion in a random binary search tree), Ch4a  116

average height (in a random increasing binary tree), Ch4a  120

average Strahler number (of a random binary tree) Ch2b-complements-Strahler, 2

aztec tilings, Ch5b  87



balanced blossoming trees, Ch2d  38

ballot paths, Ch2a  47

ballot problem, Ch2c  15, 51

basis of the Temperley-Lieb algebra, Ch2b-complements-TL_n, 18-22

Baxter permutations, Ch4b  107, Ch5a  43

Bell numbers, Ch3a  13, 54

(beta)-distribution (in the Catalan garden), Ch2c  49, Ch2c  58-61

Bernoulli numbers, Ch3b  75

bifurcation ratio (in rivers network), Ch2b-complements-Strahler, 24

bijection basis of the Temperley-Lieb algebra — staircase polygons,

                Ch2b-complements-TL_n, 23-28

bijection basis of the Temperley-Lieb algebra — (321)- avoiding permutations,

                Ch2b-complements-TL_n, 23-28

bijection binary trees — complete binary trees, Ch2a  26

bijection binary trees — (forests of) planar trees, Ch2a  36

bijection binary trees — 2-colored Motzkin paths, Ch2a  90

bijection (complete) binary trees — Dyck paths, Ch2a  65 (video with violin), 72

bijection Catalan permutations — non-crossing partitions, Ch2b  53

bijection Dyck paths — 2-colored Motzkin paths, Ch2a  55

bijection Dyck paths — Lukasiewicz paths, Ch2a  60

bijection Dyck paths — semi-pyramids of dimers, Ch2c  71 (video with violin)

bijection Hermite histories — chord diagrams, Ch4b  68-77

bijection Laguerre histories — permutations (with binary trees), Ch4b  11-22

                (with words), Ch4b  23-33,    reciprocal bijection Ch4b  34-40

bijection non-crossing partitions — Dyck paths, Ch2b  8

bijection staircase polygons — 2-colored Motzkin paths, Ch2a  105

bijection staircase polygons — Dyck paths, Ch2a  110

bijection permutations (cycles notation) — permutations (word), Ch4a  7-8, 20

bijection permutations — assemblées of increasing arborescences, Ch4a  96

bijection permutations — increasing binary trees, Ch4a  75-76

bijection permutations — inversion table,  Ch4a  29-30, 33-44

bijection permutations — inversion table (with the maj index), Ch4a  52-72

bijection permutations — pair of Young tableaux same shape (Robinson-Schensted)

                with Schensted insertions, Ch4c  24-46

                with light and shadows, Ch4c  53-73

                with growth diagrams  Ch4d  4-26

                (extension to matrices) matrices — pairs of semi-standard Young tableaux (RSK), Ch4d  63

bijection planar maps — planar quartic maps, Ch2d  18

bijection planar trees — Dyck paths, Ch2a  92

bijection planar trees — Lukasiewicz paths, Ch2a  98

bijection plane partitions — non-intersecting configuration of paths,

bijection rooted planar maps — rooted planar quartic maps, Ch2d  20

bijection rooted planar quartic maps — well balanced blossoming trees,

                (Schaeffer bijection) Ch2d  26-44

bijection aztec tilings — non-intersecting configurations of Schröder paths, Ch5b  94

bijection triangulations — (complete) binary trees, Ch2b  23 (video with violin), 31

bijection semi-standard Young tableaux — non-intersecting configurations of paths,

bijective proof  (of an identity)  Ch1a  113

bijective proof for the number of aztec tilings, Ch5b  94-113

bijective proof for the number of Baxter permutations, Ch4b  108-122, 45-48, 81-86

bijective proof of the multiplicative recursion for Catalan numbers, Ch2a  30, Ch2b  60

bijective proof for the formula giving the Catalan number Ch1a  127, 132, Ch2c  22

bijective proof of Lagrange inversion formula, Ch2c  28-39

bijective proof of Mehler identity, Ch3b  27-34

bijective proof of Touchard identity, Ch2b  73

binary search tree, Ch4a  103

                analysis of the insertion in —, Ch4a  115

binary tree, Ch2a  11

binary tree (complete),  Ch1a  3, Ch2a  10

binohedron, Ch4d-complements  56

binomial determinant, Ch5a  29

binomial power series  Ch1a  42

binomial type polynomial, Ch3b  36

bilateral Dyck path, Ch1b  42

blossoming trees, Ch1d  37

bounce parameter (q-Catalan), Ch2d 65-67

Boolean lattice, Ch4d-complements  22

Bousquet-Mélou, Mireille, Ch1d  86

Bruhat order (weak), Ch4d-complements  24



canopy (of  binary tree), Ch2d  97-98, 92

Cartier-Foata commutation monoids  Ch1b  36

Catalan numbers  Ch1a  7, Ch1b  38

                q-analogue of Catalan numbers, Ch2d  57-63

Catalan factorisation, Ch2d  76-77

Catalan permutations, Ch2b  41

Catalan triangle, Ch2c  16

Catalan word, Ch2d  78

catalytic variables, Ch1d  85

Cayley tree (species), Ch3a  15, 44-45

cellular ansatz, Ch4d  50, Ch4d-complements  18

Charlier polynomials, Ch3b-complements  44

Chomsky-Schützenberger theorem, Ch1d  75

chords diagram, Ch2b  35

circuit, Ch1b  83

cofactor, Ch1b  90

combinatorial object (class of)  Ch1a  56

complete symmetric functions, see homogeneous symmetric functions

concatenation (of two words)  Ch1a  44

conjugate words, Ch2c  4

                labelled —, Ch2c  4

contents (Young tableaux), Ch5a  54-64, 76

context-free grammar, see algebraic grammar

context-free language, see algebraic language

control theory, Ch3b-complements  13

convex polyomino, Ch1d  61—64

Cori-Vauquelin  (planar maps), Ch2d  31

covering relation (poset), Ch4d  12

crossing (of a chord diagram), Ch4b  86

crossing condition (for the LGV Lemma), Ch5a  8

cycle, Ch1b  84

cycle (species), Ch3a  12, 50

cyclic lemma, Ch2c  5 



decreasing sequence (in a permutation), Ch4c  101

« déployé" of a permutation, Ch4a  76

depth-first search (in a planar tree), Ch2a  35

derangements, Ch3a  12,  25

derivative (of formal power series)  Ch1a  41

derivative (species), Ch3a  49

derivative (L-species), Ch3b  57

derivative (weighted species), Ch3b  13

derivation tree, Ch1d  74

descent (of a permutation), Ch4a  16

De Sainte-Catherine, Ch5b  156-162

determinant, Ch1b  89

D-finite power series, Ch1b  77

differential equations (with forcing terms), Ch3b-complements  3, 13

directed animal, Ch1d  42

            random, Ch1d  47—48

            on a circular strip Ch1d  51

directed path, Ch1c  21

Dixon elliptic functions, Ch3b-complements  38

double descent (of a permutation), Ch4a  79

double rise (of a permutation), Ch4a  79

double vertex (of a binary tree), Ch2a  14

DLA (Diffusion Limited Aggregation), Ch2b-complements-Strahler, 28

D-partitions  Ch1a  74

dual (of a Young tableau), Ch4c-complement  19-52, 45

duality (for paths and non-intersecting configurations of paths), Ch5b  31, 32

Duffing equation, Ch3b-complements  20

Dyck language, Ch1d  72

Dyck lattice, Ch4d-complements  38-40

Dyck paths (definition) Ch1a  134, Ch2a  46

            (bounded) Ch1c  23



elementary circuit, Ch1b  84

elementary symmetric functions, Ch5b  15

empty species, Ch3a  18

endofunction (species), Ch3a  13, 36

Erdös-Szekeres theorem, Ch4c  105

excedance (of a permutation), Ch4a  18

exponential generating functions  Ch1a  34

exponential (of formal power series)  Ch1a  42

Euler letter to Goldbach (introducing Catalan numbers),  Ch1a  15, Ch2b  17-20

Euler formula for convex polyhedron, Ch2b  21

Euler-Descartes defects formula for polyhedron, Ch2b  21-22

Euler pentagonal theorem, Ch1c  77, Ch1d  4—13

Eulerian polynomials, Ch4a  17, 19

exponential polynomial, Ch3b  39

external vertex  Ch1a  6



factor  (left, right) of a word  Ch1a  45

factorisation  (of a word)  Ch1a  45

Favard theorem, Ch4b  44

Fermat matrix (of binomial coefficients), Ch5b  38

            inverse matrix, 39-41

Ferrers diagram,  Ch1a  64

Fibonacci generating function  Ch1a  23

Fibonacci polynomials, Ch1c  30, 35, 44, Ch1d  15

Flajolet, Philippe, Ch3b-complements 38, Ch2b-complements-Strahler 2

Fliess expansion, Ch3b-complements  4

Foata, Dominique, Ch4a  17, 102

Fomin, Sergey, Ch4d  3, 27, Ch4d  45

forest of planar trees, Ch2a  33

free monoid,  Ch1a  44

forbidden pattern (for permutations), Ch2b  54

formal power series  Ch1a  32

Françon, Jean, Ch1b  64, Ch2b  116

Frobenius identity, Ch4a  17



galactogram (in radiology), Ch2b-complements-Strahler, 30

(gamma)-distribution (in the Catalan garden), Ch2c  62-64

generating function (of a class of combinatorial objects)  Ch1a  57

generating function (of a species), Ch3a  9

generating functions (of weigthed species), Ch3b  8

Genocchi numbers, Ch3b  75

Gessel path, 84

growth diagrams, Ch4d  3



Hankel determinant, Ch5b  55

            — of Catalan numbers, Ch5b  153-154

Hasse diagram, Ch1d  14, Ch4d  12

heap of dimers, Ch1b 21

            generating function, Ch1d  15

            connected, Ch1d  87

height, left-, right- (of a vertex in a binary tree), Ch2a  15

Hermite polynomials, Ch1c  62, Ch3b  15-19, Ch3b-complements  42

            moments of —, Ch4b  69-71

            q-analogue I, Ch4b  79-86, q-analogue II, Ch4b  100

Hermite histories, Ch4b  68-77, Ch4d  54

            q-analogue I, Ch4b  79-86,  q-analogue II, Ch4b  100

Hipparcus number, Ch5b  121-124

histories (differential equations), Ch3b-complements  17

holonomic power series, see D-finite or P-recursive

homogeneous symmetric functions, Ch5b  14

hook length formula (for increasing binary trees), Ch4a  90

            (for Young tableaux), Ch4c  8, Ch5a  54-64, 76

Hydrogeology, Ch1b  49, Ch2b-complements-Strahler, 24



increasing binary trees, Ch3b  44, Ch4a  74

infinite product (of formal power series)  Ch1a  38

integral (of an L-species), Ch3b  62

iterated integral, Ch3b-complements  5, 34

increasing sequence (in a permutation), Ch4c  98

internal vertex  Ch1a  6

inorder (in a binary tree), 21

inverse (of formal power series) Ch1a  41

inversions number (of a permutation), Ch4a  26 

inversion table (of a permutation), Ch4a  26 

involution (species), Ch3a  11, 33

involution with no fixed points (species), Ch3a  11, 33

irreducible representation (of the symmetric group), Ch4c  86

Ising model, Ch5b  150-151

isolated point (of a matching) Ch1c  58



Jacobi elliptic functions, Ch3b  79, Ch3b-complements  36

Jacobi-Trudi identities (for Schur functions), Ch5b  12

            (for skew Schur functions) 26, 28 

Jacobi permutations, Ch4a  98 

Jeu de taquin, Ch4c-complement  2-8



Kastelyn formula, Ch5b  66-67, 126-129

            Kastelyn theorem, Ch5b  145

kernel methodology, Ch1d  85

Knuth, Don, Ch1b  36, Ch2b  116, Ch4c  51, Ch4c-complement  55

Knuth transpositions, Ch4c-complement  55

Kreweras path, Ch1d  84

            Kreweras determinants, Ch5b  49



lacet, see circuit

Lagrange inversion formula, Ch1d  22—23, 32, Ch2c  27-39

Laguerre configuration, Ch3b  23

Laguerre histories, Ch4b  4-10

            weighted —, Ch4b  54

            restricted —, Ch4b  60

            q-analogues I, Ch4b  91-94, Ch4d-complements  14-16

            q-analogue II, Ch4b  96

Laguerre polynomials, Ch1c  63, Ch3b  20-25

            moments of —, Ch4b  48-53, 59, 66

            q-analogue I, Ch4b  91-94, 98, Ch4d-complements  14-16

            q-analogue II, Ch4b  96

Lah numbers, Ch3b  6

langage  (in a free monoid) Ch1a  45

Lascoux, Alain, Ch4c-complement  60, Ch4d-complements  53, Ch5a  37

lattice, Ch4d-complements  20

leaf  Ch1a  6

length (of a word)  Ch1a  44

Leroux, Pierre, Ch3b-complements  13

LGV Lemma, Ch5a  3, 23

linear species, Ch3b  41

Loday, Jean-Louis, Ch4d-complements  46-48

logarithm  (of formal power series)  Ch1a  42

logarithmic height (of a Dyck path), Ch1c  68, Ch2b  103

logarithmic lemma  (for heaps of dimers), Ch1b  26

Lucas polynomials, Ch1c  50

Laplace, Ch3b-complements  37

Lukasiewicz path, Ch2a  51

Lukasiewicz word, Ch2a  52



MacMahon, Percy, Ch4a  51

            MacMahon formula for plane partitions in a box, Ch5a  96, 106-110

            Macmahon-Narayana determinant  Ch5b  42

Mahonian distribution, Ch4a  51

Maj index (q-Catalan number), Ch2d 57

Maj index (permutations), Ch4a  51, 52-72

Markov chain, Ch2d  91

matching (of the segment graph) Ch1c  30

matching polynomial (of a graph) Ch1c  57

Mehler identity (for Hermite polynomials), Ch3b  27

Meixner I, II polynomials, Ch3b-complements  44

minimum elements (left-to-right), Ch4a  9

moments (of orthogonal polynomials), Ch3b-complements  41, Ch4b  47

Motzkin path, Ch1d  46, Ch2a  48

            (prefix of) Ch1d  45

            2-colored —, Ch2a  50



Narayana numbers, Ch2c  51, Ch5a  44, 83

nesting (of a chord diagram), Ch4b  84

neutral element (in a free monoid) Ch1a  44

Nim game (on the Rothe diagram of a permutation), Ch4c  79 

non-commutative formal power series  Ch1a  46

non-crossing partitions, Ch2a  118, 122, Ch2b  7



ordered trees, see planar trees

operations of combinatorial objects  Ch1a  47, 55

operations on species, Ch3a  22

operations on L-species, Ch3b  55

operations on weighted species, Ch3b  9

operators U and D (combinatorial representation), Ch4d   45

oscillating tableaux, Ch4d  51

            2-colored —  Ch4d  56

orthogonal polynomials (formal), Ch3b-complements  40

orthogonal Sheffer polynomials, Ch3b-complements  44

outstanding elements (permutations), Ch4a  15



partitions (of integers)  Ch1a  64

partitions (species), Ch3a  13, 36

PASEP, Ch4d-complements  2

Pascal triangle, Ch1b  60

Path, Ch1b  81

            generating function N/D, Ch1b  86

            bijective proof for N/D,  Ch1c  9

            in a quadrant, Ch1d  83

path length (of a binary tree), Ch1d  86

pattern (31-2) in a permutation, Ch4b  40

peak (of a permutation), Ch4a  79

peak (of a Dyck path),

pebbles problem on a binary tree, Ch2b-complements-Strahler, 7-23

perfect matching, Ch1c  58

            and tilings, Ch5b  79

permutahedron, Ch4d-complements  53-54

permutations, Ch1b  85, Ch4a  3

            (graphical representation), Ch4a  5

permutations sortable by one stack, see Catalan permutations

permutations tableaux, Ch4d-complements  11

permutations, (312)-avoiding  — , Ch2b  55

Pfaffian (definition), Ch5b  146

            Pfaffians methodology, Ch5b  148

Pingala, Ch1b  62, Ch1c  41

plactic monoid, Ch4c-complement  60

planar maps, Ch1d  26, Ch2d  10

planar trees, Ch2a  32

plane partitions, Ch5a  92

pointed combinatorial object, Ch1b 17

pointed species, Ch3a  43

pointed weighted species, Ch3b  12

Polya q-analogue, Ch1d  63

Polya urn model, Ch3b-complements  37

polyomino, Ch1d  55

            convex Ch1d  61—64

            random convex Ch1d  66—68

poset (species), Ch3a  14, Ch4d  12

postorder (in a binary tree), Ch2a  23

P-recursive power series, Ch1b  77

preorder (in a binary tree), Ch2a  17

preorder (in a planar tree), Ch2a  35

primitive word, Ch2c  4

principal branch, left-, right- (of a binary tree), Ch2a  16

product (of combinatorial objects)  Ch1a  60

product (of species), Ch3a  24

product (of weighted species), Ch3b  10

pyramids (of dimers) Ch1b  26

Pythagoras theorem (visual proof)  Ch1a  65



q-analogues (introduction), Ch2d 54-56, Ch4a  25

q-analogues of Catalan numbers, Ch2d  57-63

(q,t)-Catalan, Ch2d  65-74

quantum gravity, Ch1d  40

quartic planar maps, Ch2d  18



Ramanujan  Ch1a  77

ramification matrix (of a binary tree), Ch2b-complements-Strahler, 26

random directed animal, Ch1d  47-48

random binary tree, Ch2b-complements-Strahler, 42, 44

random binary search tree, Ch2b-complements-Strahler, 40, 44

random convex polyominoes, Ch1d  66—68

rational language, Ch1d  77

rational power series, Ch1b  77, 80, Ch1d  17

reflection principle (the), Ch2c  41-47, 49-53

registers (minimum number for computing an arithmetical expression),

            Ch2b-complements-Strahler, 4-6

representation theory of groups, Ch4c  82

rise (of a permutation), Ch4a  16

Rogers-Ramanujan identities  Ch1a  79

rook placements, Ch4d  53, 56, 59

Rothe diagram, Ch4a  28

RSK, (Robinson-Schensted-Knuth) introduction Ch4c  17, Ch4d  63

            with Schensted insertion, Ch4c  23

            geometric version with light and shadow, Ch4c  52



Schaeffer bijection (for planar maps), Ch1d  34, Ch2d  25-44

Schensted (bumping process), Ch4c  23

Schröder numbers, Ch5b  114

Schur functions, Ch4c-complement  68, Ch5b  16

            skew —, Ch5b  23

Schützenberger, Marcel-Paul, Ch3b  78, Ch4a  17, 102, Ch4c-complement  2, 9, 60

Schützenberger methodology, Ch1d  31, 70

secant numbers, Ch3b  56, 71, Ch4a  22, 97

self-avoiding path, Ch1b  83

separation of variables (differential equation), Ch3b-complements  24-28

sequence (of combinatorial objects) Ch1a  63

semi-pyramids of dimers, Ch1b  38, Ch2c  66-69

Sheffer polynomials, Ch3b 36

shuffle product, Ch3b-complements  7

simple vertex, right, left (of a binary tree), Ch2a  14

singleton (species), Ch3a  18

species, Ch3a  4

skeleton (of a permutation), Ch4c  65, 77

skew Schur functions, Ch5b  23

stack polyomino, Ch1c  43

staircase polygon, Ch2a  104

stair decomposition if a heap of dimers, Ch2b-complements-TL_n, 15

stationnary probabilities, Ch1d  93

            for the TASEP, Ch2d  96, 99

            for the PASEP, Ch4d-complements  10

Stirling numbers 1st kind, Ch3b  38, Ch4a  11

Stirling numbers 2nd kind, Ch3b  39

Strahler number (of a binary tree), Ch1b  47, Ch1c  64, Ch2b  106-118

            Ch2b-complements-Strahler, 2-50

sub-excedante function, Ch4a  24

substitution (in formal power series)  Ch1a  40

substitution (for combinatorial objects), Ch1b  45

substitution (of species), Ch1d  28

substitution (of weighted species), Ch3b  11

subtree, left, right (of a binary tree), Ch2a  13

subword  Ch1a  45

sum (of combinatorial objects) Ch1a  59

sum (of species), Ch1d  23

sum (of weigthed species), Ch3b  9

summable family of formal power series  Ch1a  35

symmetric functions, Ch5b  13

            elementary —, Ch5b  15

            homogeneous (complete) — Ch5b  14

symmetric group, Ch4a  6

symmetric order (in a binary tree), see inorder

synthetic images of trees and landscapes, Ch2b-complements-Strahler, 33-50



Tamari lattice, Ch4d-complements  26

tangent numbers, Ch3b  65, 71, Ch4a  22, 97, Ch5a  90

TASEP, Ch2d  90-103, Ch5b  42

Tchebychef polynomials 2nd kind, Ch1c  44

Tchebychef polynomials 1st kind, Ch1c  51

Temperley-Lieg algebra, Ch2b-complements-TL_n, 2-34

tilings, Ch5b  63

            aztec, Ch5b  87

            on a rectangle, Ch5b  65-68, 125-129

            on a triangular lattice, Ch5b  69

Touchard identity, Ch2a  59, Ch2b  73

trace monoids  Ch1b  36

transition matrix methodology, Ch1b  86

transport (of structures), Ch3a  6

triangulation of  a convex polygon  Ch1a  14, Ch2b  14

trough (of a permutation), see valley

Tutte, Ch1d  28



uniform species, Ch3a  17

up-down sequence (of a permutation), Ch4a  21, 92, Ch5a  89



valley (or a permutation), Ch4a  79

Vandermonde determinant, Ch5a  66-67 

vertébré, Ch3a  46

vertically convex polyomino, Ch1d  59

viscous fingering, Ch2b-complements-Strahler, 29



Walks, see paths

weight (of a heap of dimers) Ch1b 24

weighted species, Ch3b  3

    — L-species, Ch3b  69

well balanced blossoming trees, Ch2d  42



x-factorisation (of a permutation), Ch4a  77

x-decomposition (of a permutation), Ch4b  36



Young lattice, Ch4d  15

Young tableaux, Ch4c  3

            semi-strandad —, Ch5a  68 


zeros (of matching polynomial of graphs), Ch1c  60

            (of Fibonacci polynomials) Ch1d  1

            (of Lucas polynomials) Ch1d  20




last update: 6 April 2016


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