Index

<_lex, 136

λ ⊢ n, 113

τ, 22

, 109

Alder, H. L., 34, 38

Alder’s conjecture, 34

Andrews, G. E., 12, 33, 40, 35, 61, 123

Andrews’s identity, 33

arctic circle, 119, 120

arctic circle theorem, 120

arctic zone, 119

Artin conjecture, 123

Atkin, A. O. L., 51

Aztec diamond, 116

   number of tilings, 116

bad block, 117

Beal, A., 40

bijection, 7

bijective method, 5, 7

binary representation of integer, 10

binomial number, 64

binomial series, 65

binomial theorem, 65

Bonaccio, 20

Borwein’s problem, 125

Bousquet-Mélou, M., 93

Bressoud, D., 23, 36, 41, 81

cardinality, 6

Catalan number, 22, 113

Chung, F., 40

composition of integer, 21

congruence modulo 5, 32

conjugation, 18, 110

Coxeter group, 92

d-distinct parts, 31

dimer, 115

disjoint sets, 6

distinct odd parts, 18

distinct part, 8

distinct parts, 17

dominance order, 109

domino, 115

domino tiling

   of chessboard, 115

   of Ferrers board, 136

   of staircase, 116

Durfee square, 19, 76

   successive, 86

Dyson, F., 91

element, 5

Elkies, N., 117

Erdös, P., 40

Eriksson, K., 93

Euler identity, 3, 6, 31

   Alder-type formulation, 34

   analytic proof, 47

   bijective proof, 9

   Fine’s refinement, 91

   finite version, 94

   lecture hall refinement, 92

   Sylvester’s refinement, 88

Euler pair, 11

   and Schur’s theorem, 36

Euler pair theorem, 11

Euler’s pentagonal number theorem, 25, 81

   Franklin’s bijective proof, 27

   with generating functions, 50

Euler, L., 2, 5, 24

f^λ, 112

Ferrers board, 15

   Durfee square of, 19

   partial order of, 108

Ferrers graph, 15

Fibonacci number, 20

   growth of, 22

Fine, N., 28, 91

Fisher, M. E., 115

fixed-point, 17

Franklin, F., 25

Frobenius symbol, 78

Garsia, A., 41

Gaussian formula

   for Gaussian polynomials, 71

Gaussian polynomial, 71, 123

   limiting value, 74

golden mean, 22

good block, 117

Göllnitz-Gordon identity

   first, 33, 38

   second, 33

Gordon’s identity, 33

Graham, R., 40

growing function, 20

Hall, M., 92

Hall, P., 92

Hardy, G. H., 61

Hasse diagram, 108

hook, 111

hook length formula, 111, 112

inner corner, 107

inner corner of a Ferrers board, 16

integer partition, 2

   into 2-distinct parts, 31

   representation, 15

intersection of sets, 6

involution principle, 41

Jacobi’s triple product identity, 80

Jockush, W., 114, 120

Kasteleyn, P. W., 115

Kuperberg, G., 117

Larsen, M., 117

lattice path, 67

lecture hall partition, 92

Lehmer, D. H., 34

Leibniz’s problem, 125

Leonardo of Pisa, 20

lexicographic order, 136

Liber Abaci, 20

long rectangle, 18

MacMahon’s plane partition formula, 101

MacMahon, P. A., 48, 51, 101

Mann, H., 123

merging two parts, 7

merging/splitting process, 8, 11

Milne, S., 41

modular arithmetic, 3

number of parts, 17

odd part, 8

Omega, 123

outer corner of a Ferrers board, 16

parity of p(n), 125

partial order, 108

partition function, 6

   congruences, 51

   rapid computation, 50

   table, 51

   upper bound, 19, 22

partition identity, 3

   bijective proof, 7

   fundamental structure of, 29

   notation, 6

Pascal’s triangle, 65

Paule, P., 123

pentagonal number, 25

pentagonal number theorem, 25

petroglyph, 1

p(n | [condition] ), 6

p(n), 6, 19, 22

poset, 108

proof by induction, 22

Propp, J., 114, 117, 119

Putnam examination, 124

q-analog, 67

q-binomial number, 67

q-binomial series, 70

q-binomial theorem, 70

Rademacher, H., 61

Ramanujan, S., 3, 32, 61

random growth, 107

   type I, 113

   type II, 114, 120

random partition, 106

rank

   of a partition, 90

recursive definition

   of Catalan numbers, 22

   of Fibonacci numbers, 20

remainder, 3

Riese, A., 123

rim of a Ferrers board, 16

Robinson-Schensted-Knuth correspondence, 136

Rogers, L. J., 3

Rogers-Ramanujan identities

   generating functions, 52

   proof of, 81

   verification, 32

Rogers-Ramanujan identity

   Alder-type formulation, 34

   first, 3

   looking for a bijective proof, 39

Rost, H., 113, 115

row insertion, 110

Schur, I., 12, 32, 35

Schur’s identity, 33

Schur’s theorem, 35

self-conjugate, 17

set, 5

set theory, 5

Shor, P., 114, 120

shuffling, 117

sizes of parts, 17

splitting a part, 7

staircase, 112

   height, 22

   number of Ferrers boards contained in, 22

standard tableau, 110

   number of, 112

   shape of, 111

standard Young tableau, 110

statistical physics, 115

Subbarao, 48

subexponential function, 61

subset, 6

super-distinct, 23

Sylvester, J.J., 28, 88

temperate zone, 119

Temperley, H. N. V, 115

tile, 115

tiling, 115

triangular number, 25

union of sets, 6

usual order, 108

Venn diagram, 6

Watson, G. N., 51

Young diagram, 15

Zeilberger, D., 41