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A N A L Y T I C C O M B I N A T O R I C S P A R T O N E 5. Analytic Combinatorics http://aofa.cs.princeton.edu

Analytic combinatorics is a calculus for the quantitative study of large combinatorial structures. Features : • Analysis begins with formal combinatorial constructions . • The generating function is the central object of study. • Transfer theorems can immediately provide results from formal descriptions. • Results extend, in principle, to any desired precision on the standard scale. • Variations on fundamental constructions are easily handled. generating symbolic analytic coefficient combinatorial function transfer transfer asymptotics constructions equation theorem theorem the “symbolic method” 2

Analytic combinatorics is a calculus for the quantitative study of large combinatorial structures. Ex : How many binary trees with N nodes? � � � ( � ) = � + �� ( � ) � T = E + Z × T × T � � ∼ � �� � combinatorial construction GF equation coefficient asymptotics 3

A N A L Y T I C C O M B I N A T O R I C S P A R T O N E 5. Analytic Combinatorics •The symbolic method •Labelled objects •Coefficient asymptotics OF •Perspective http://aofa.cs.princeton.edu 5a.AC.Symbolic

The symbolic method is an approach for translating combinatorial constructions to GF equations Examples A, B, Z • Define a class of combinatorial objects. | b | • Define a notion of size. A ( z ) • Define a GF whose coefficients count objects of the same size. A × B • Define operations suitable for constructive definitions of objects. • Develop translations from constructions to operations on GFs. A ( z ) B ( z ) Building blocks Building blocks Building blocks Formal basis : • A combinatorial class is a set of objects and a size function . notation denotes contains • An atom is an object of size 1 . atomic Z an atom class • An neutral object is an atom of size 0 . neutral neutral E • A combinatorial construction uses the union, product, and class object sequence operations to define a class in terms of atoms empty and other classes. Φ nothing class 5

Unlabelled class example 1: natural numbers Def. A natural number is a set (or a sequence) of atoms. counting sequence OGF � ● ●● ●●● ●●●● ●●●●● � � = � � − � I 2 = 1 I 4 = 1 I 5 = 1 I 1 = 1 I 3 = 1 � � � = � � − � � ≥ � unary notation 6

Unlabelled class example 2: bitstrings Def. A bitstring is a sequence of 0 or 1 bits. 0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 0 1 0 0 0 0 0 0 1 0 1 0 0 1 counting sequence OGF 0 1 1 0 0 0 0 1 0 � 0 0 1 1 1 0 1 0 1 1 � � = � � � − � � 1 0 0 0 1 1 0 1 0 0 1 0 0 1 1 1 1 0 1 B 0 = 1 B 1 = 2 1 0 1 0 1 1 0 � � � � � = ( � � ) � = � � B 2 = 4 1 0 1 1 1 1 1 � − � � � ≥ � � ≥ � 1 1 0 0 B 3 = 8 1 1 0 1 1 1 1 0 1 1 1 1 B 4 = 16 7

Unlabelled class example 3: binary trees Def. A binary tree is empty or a sequence of a node and two binary trees counting sequence OGF � � � � � � √ � � = � � ( � − � − � � ) � + � � T 1 = 1 T 2 = 2 Catalan numbers (see Lecture 3) � ( � ) = � + �� ( � ) � T 3 = 5 T 4 = 14 8

Combinatorial constructions for unlabelled classes construction notation semantics A + B disjoint copies of objects from A and B disjoint union A and B are ordered pairs of copies of objects, A × B combinatorial classes Cartesian product one from A and one from B of unlabelled objects SEQ ( A ) sequences of objects from A sequence ( + ) × ( + + ) = Ex 1. 0 0 0 1 1 0 1 1 1 0 1 1 1 0 0 1 0 1 0 0 1 1 0 0 0 1 1 1 0 1 1 0 1 0 1 1 1 1 ● × SEQ ( ● ) = ● ●● ●●● ●●●● ●●●●● ●●●●●● ●●●●●●● ... Ex 2. □ × ● × ● = ● Ex 3. "unlabelled" ?? Stay tuned. ● □ □ □ □ □ 9

The symbolic method for unlabelled classes (transfer theorem) Theorem. Let A and B be combinatorial classes of unlabelled objects with OGFs A ( z ) and B ( z ). Then construction notation semantics OGF � ( � ) + � ( � ) A + B disjoint copies of objects from A and B disjoint union ordered pairs of copies of objects, � ( � ) � ( � ) A × B Cartesian product one from A and one from B � SEQ ( A ) sequences of objects from A sequence � − � ( � ) 10

Proofs of transfers are immediate from GF counting A + B � | γ | = � | α | + � | β | = � ( � ) + � ( � ) � � � γ ∈ � + � α ∈ � β ∈ � A × B � | γ | = � | α | + | β | = �� � | α | ��� � | β | � � � � = � ( � ) � ( � ) γ ∈ � × � α ∈ � β ∈ � α ∈ � β ∈ � SEQ ( A ) ��� ( � ) ≡ � + � + � � + � � + � � + . . . � � + � ( � ) + � ( � ) � + � ( � ) � + � ( � ) � + . . . = � − � ( � ) 11

Symbolic method: binary trees How many binary trees with N nodes? Atoms type class size GF Class T , the class of all binary trees � � external node 0 1 Size | t |, the number of internal nodes in t � • 1 z internal node � | � | � � � � � � ( � ) = � = OGF � ≥ � � ∈ � “a binary tree is an external node or an internal node connected to two binary trees” � = � � + � × � • × � Construction or � ( � ) = � + �� ( � ) � OGF equation � � � � � � � [ � � ] � ( � ) = ∼ � + � � � �� � see Lecture 3 and stay tuned. 12

Symbolic method: binary trees How many binary trees with N external nodes? Atoms Class T , the class of all binary trees type class size GF � � external node 1 z Size ☐ , the number of external nodes in t t � • 0 1 internal node � � ( � ) = � � � t OGF � ∈ � “a binary tree is an external node or an internal node connected to two binary trees” � = � � + � × � • × � Construction or � � ( � ) = � + � � ( � ) � OGF equation � � ( � ) = �� ( � ) [ � � ] � � ( � ) = [ � � − � ] � ( � ) = � � � � − � � same as # binary trees � � − � with N − 1 internal nodes 13

Symbolic method: binary strings Warmup: How many binary strings with N bits? Atoms type class size GF Class B , the class of all binary strings � � 0 bit 1 z Size | b |, the number of bits in b � � 1 z 1 bit � | � | � � � � � � ( � ) = � OGF = � ≥ � � ∈ � “a binary string is a sequence � = ��� ( � � + � � ) Construction of 0 bits and 1 bits” � � ( � ) = OGF equation � − � � [ � � ] � ( � ) = � � ✓ 14

Symbolic method: binary strings (alternate) Warmup: How many binary strings with N bits? Atoms type class size GF Class B , the class of all binary strings � � 0 bit 1 z Size | b |, the number of bits in b � � 1 z 1 bit � | � | � � � � � � ( � ) = � OGF = � ≥ � � ∈ � “a binary string is empty or � = � + ( � � + � � ) × � Construction a bit followed by a binary string” � ( � ) = � + � �� ( � ) OGF equation � � ( � ) = Solution � − � � [ � � ] � ( � ) = � � ✓ 15

Symbolic method: binary strings with restrictions Ex. How many N -bit binary strings have no two consecutive 0s? 0 1 0 1 1 0 1 0 1 0 0 1 1 0 1 0 1 0 1 0 1 1 1 0 0 1 1 0 0 1 1 0 1 1 1 1 0 1 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 1 1 0 1 0 1 1 0 1 1 0 1 1 0 1 0 1 1 1 1 0 1 0 1 1 1 1 1 0 1 T 0 = 1 T 1 = 2 1 1 1 1 1 0 1 1 T 2 = 3 1 1 1 0 1 1 0 1 0 T 3 = 5 1 1 1 1 1 1 1 0 1 1 1 1 1 0 T 4 = 8 1 1 1 1 1 T 5 =13 Stay tuned for general treatment (Chapter 8) 16

Symbolic method: binary strings with restrictions Ex. How many N -bit binary strings have no two consecutive 0s? Atoms type class size GF Class B 00 , the class of binary strings with no 00 � � 0 bit 1 z Size | b |, the number of bits in b � � 1 z 1 bit � | � | � �� ( � ) = � OGF � ∈ � �� “a binary string with no 00 is either � �� = � + � � + ( � � + � � × � � ) × � �� Construction empty or 0 or it is 1 or 01 followed by a binary string with no 00” � �� ( � ) = � + � + ( � + � � ) � �� ( � ) OGF equation � + � � �� ( � ) = solution � − � − � � [ � � ] � �� ( � ) = � � + � � + � = � � + � ✓ 1, 2, 5, 8, 13, ... 17

Symbolic method: many, many examples to follow How many ... with ... ? Atoms type class size GF Class Size OGF “a ... is either ... Construction or ... and ...” OGF equation solution 18

A N A L Y T I C C O M B I N A T O R I C S P A R T O N E 5. Analytic Combinatorics •The symbolic method •Labelled objects •Coefficient asymptotics OF •Perspective http://aofa.cs.princeton.edu 5a.AC.Symbolic

A N A L Y T I C C O M B I N A T O R I C S P A R T O N E 5. Analytic Combinatorics •The symbolic method •Labelled objects •Coefficient asymptotics OF •Perspective http://aofa.cs.princeton.edu 5b.AC.Labelled