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Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Catalan Numbers Mike Joseph University of Connecticut December 5, 2014 Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Ways to Insert Parentheses in Multiplication of n Elements Suppose we have multiplication of n elements. Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Ways to Insert Parentheses in Multiplication of n Elements Suppose we have multiplication of n elements. Examples: abc or abcd. Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Ways to Insert Parentheses in Multiplication of n Elements Suppose we have multiplication of n elements. Examples: abc or abcd. How many ways can we insert parentheses so that only two elements are being multiplied at once? Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Ways to Insert Parentheses in Multiplication of n Elements For n = 3, there are 2 ways: (ab)c and a(bc). Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Ways to Insert Parentheses in Multiplication of n Elements For n = 3, there are 2 ways: (ab)c and a(bc). What about n = 4? What are the ways to parenthesize abcd? Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Ways to Insert Parentheses in Multiplication of n Elements For n = 3, there are 2 ways: (ab)c and a(bc). What about n = 4? What are the ways to parenthesize abcd? ((ab)c)d Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Ways to Insert Parentheses in Multiplication of n Elements For n = 3, there are 2 ways: (ab)c and a(bc). What about n = 4? What are the ways to parenthesize abcd? ((ab)c)d a(b(cd)) Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Ways to Insert Parentheses in Multiplication of n Elements For n = 3, there are 2 ways: (ab)c and a(bc). What about n = 4? What are the ways to parenthesize abcd? ((ab)c)d a(b(cd)) (ab)(cd) Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Ways to Insert Parentheses in Multiplication of n Elements For n = 3, there are 2 ways: (ab)c and a(bc). What about n = 4? What are the ways to parenthesize abcd? ((ab)c)d a(b(cd)) (ab)(cd) (a(bc))d a((bc)d) There are 5 ways. Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Ways to Insert Parentheses in Multiplication of n Elements For n = 5, there are 14 ways. a(((bc)d)e) a(b(c(de))) a((bc)(de)) a((b(cd))e) a(b((cd)e)) (ab)((cd)e) (ab)(c(de)) ((ab)c)(de) (a(bc))(de) (((ab)c)d)e (a(b(cd)))e ((ab)(cd))e ((a(bc))d)e (a((bc)d))e Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Ways to Insert Parentheses in Multiplication of n Elements n Parenthesizations 1 1 2 1 3 2 4 5 5 14 Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Polygon Triangulations Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Polygon Triangulations Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Polygon Triangulations n 3 4 5 6 Triangulations of convex n-gon 1 2 5 14 Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Balls into Boxes Count the number of ways to put n indistinguishable balls into n boxes labeled B1 , B2 , . . . , Bn such that the collection B1 , . . . , Bi has at most i balls, for any i ∈ {1, . . . , n}. Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Balls into Boxes Count the number of ways to put n indistinguishable balls into n boxes labeled B1 , B2 , . . . , Bn such that the collection B1 , . . . , Bi has at most i balls, for any i ∈ {1, . . . , n}. For example, if n = 3, then these are the ways to place 3 balls into boxes B1 , B2 , B3 such that B1 has at most one ball. B1 , B2 together have at most two balls. Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Balls into Boxes For n = 3, these are the possible distributions of balls into boxes. These are listed as ordered triples (#B1 , #B2 , #B3 ). (0, 0, 3) (0, 1, 2) (0, 2, 1) (1, 0, 2) (1, 1, 1) Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Balls into Boxes For n = 3, these are the possible distributions of balls into boxes. These are listed as ordered triples (#B1 , #B2 , #B3 ). (0, 0, 3) (0, 1, 2) (0, 2, 1) (1, 0, 2) (1, 1, 1) There are 5 distributions. Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Balls into Boxes For n = 1, there is a unique distribution (1). Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Balls into Boxes For n = 1, there is a unique distribution (1). For n = 2, there are two distributions: (0, 2) and (1, 1). Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Balls into Boxes For n = 1, there is a unique distribution (1). For n = 2, there are two distributions: (0, 2) and (1, 1). For n = 4, (0, 0, 0, 4) (0, 0, 1, 3) (0, 0, 2, 2) (0, 0, 3, 1) (0, 1, 0, 3) (0, 1, 1, 2) (0, 1, 2, 1) (0, 2, 0, 2) (0, 2, 1, 1) (1, 0, 0, 3) (1, 0, 1, 2) (1, 0, 2, 1) (1, 1, 0, 2) (1, 1, 1, 1) There are 14 distributions! Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Balls into Boxes n Distributions 0 1 1 1 2 2 3 5 4 14 Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References What is the Sequence of Catalan Numbers? Sequence Cn of numbers 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, . . . Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References What is the Sequence of Catalan Numbers? Sequence Cn of numbers 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, . . . Note that the first term of the sequence is C0 , so C0 = 1, C1 = 1, C2 = 2, C3 = 5, etc. Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References What is the Sequence of Catalan Numbers? Sequence Cn of numbers 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, . . . Note that the first term of the sequence is C0 , so C0 = 1, C1 = 1, C2 = 2, C3 = 5, etc. n P Recurrence relation: Cn+1 = Ck Cn−k k=0 Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References What is the Sequence of Catalan Numbers? Sequence Cn of numbers 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, . . . Note that the first term of the sequence is C0 , so C0 = 1, C1 = 1, C2 = 2, C3 = 5, etc. n P Recurrence relation: Cn+1 = Ck Cn−k k=0 Explicit formulas: (2n)! 2n 1 = Cn = n+1 = n!(n+1)! n Mike Joseph 2n n − 2n n+1 Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References What is the Sequence of Catalan Numbers? Sequence Cn of numbers 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, . . . Note that the first term of the sequence is C0 , so C0 = 1, C1 = 1, C2 = 2, C3 = 5, etc. n P Recurrence relation: Cn+1 = Ck Cn−k k=0 Explicit formulas: (2n)! 2n 1 = Cn = n+1 = n!(n+1)! n 2n n − 2n n+1 Another recurrence relation: Cn+1 = Mike Joseph Catalan Numbers 4n+2 C n+2 n Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References What is the Sequence of Catalan Numbers? The initial condition C0 = 1 and recurrence n P Cn+1 = Ck Cn−k fully define the Catalan numbers. k=0 To prove that a set of objects are counted by Catalan numbers, we usually show that they satisfy this initial condition and recurrence relation. Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References What is the Sequence of Catalan Numbers? The initial condition C0 = 1 and recurrence n P Cn+1 = Ck Cn−k fully define the Catalan numbers. k=0 To prove that a set of objects are counted by Catalan numbers, we usually show that they satisfy this initial condition and recurrence relation. To prove that the Catalan numbers have the explicit 2n 1 formula Cn = n+1 , generating functions are used. n Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Generating Function f (x) = ∞ X Cn x n n=0 Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Generating Function f (x) = ∞ X Cn x n n=0 Property of generating functions: ! ∞ ! ∞ ∞ X X X n n hn xn an x bn x = n=0 where hn = n P n=0 n=0 ak bn−k . k=0 Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Generating Function f (x) = ∞ X Cn x n n=0 ⇒ f (x)2 = n ∞ X X n=0 Mike Joseph ! Ck Cn−k k=0 Catalan Numbers xn Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Generating Function f (x) = ∞ X Cn x n n=0 ⇒ f (x)2 = n ∞ X X n=0 = ∞ X ! Ck Cn−k k=0 Cn+1 xn n=0 Mike Joseph Catalan Numbers xn Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Generating Function f (x) = ∞ X Cn x n n=0 ⇒ f (x)2 = n ∞ X X n=0 = ∞ X ! Ck Cn−k k=0 Cn+1 xn n=0 = f (x) − 1 x Mike Joseph Catalan Numbers xn Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Generating Function After some work... f (x) = ∞ X n Cn x = 1− √ 1 − 4x 2x n=0 From this, we can prove Cn = Mike Joseph 2n 1 n+1 n . Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References History and Importance Sharabiin Myangat (c. 1692 - c. 1763), a Mongolian astronomer and mathematician, worked at the Qing court in China. He came across these numbers in the following identity. (1730s) sin(2α) = 2 sin α − ∞ X Cn−1 n=1 Mike Joseph 4n−1 Catalan Numbers sin2n+1 α Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References History and Importance Euler conjectured (1751) the number of triangulations of a convex n-gon to be Cn−2 . Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References History and Importance Euler conjectured (1751) the number of triangulations of a convex n-gon to be Cn−2 . Goldbach and Segner (1758-1759) helped Euler complete the proof, and the first self-contained complete proof was in 1838 by Lamé. Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References History and Importance Euler conjectured (1751) the number of triangulations of a convex n-gon to be Cn−2 . Goldbach and Segner (1758-1759) helped Euler complete the proof, and the first self-contained complete proof was in 1838 by Lamé. (2n)! Eugéne Catalan (1838) wrote Cn in the form n!(n+1)! and showed it to be the number of ways to parenthesize multiplication of n + 1 elements. Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References History and Importance Riordan introduced the term "Catalan numbers" in Math Reviews (1948). Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References History and Importance Riordan introduced the term "Catalan numbers" in Math Reviews (1948). The name caught on after Riordan used it in Combinatorial Identities (1968). Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References History and Importance Riordan introduced the term "Catalan numbers" in Math Reviews (1948). The name caught on after Riordan used it in Combinatorial Identities (1968). Richard Stanley has compiled a list of 214 combinatorial interpretations of the Catalan numbers, which will appear along with 68 additional problems in a monograph to be published in 2015. Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References History and Importance Riordan introduced the term "Catalan numbers" in Math Reviews (1948). The name caught on after Riordan used it in Combinatorial Identities (1968). Richard Stanley has compiled a list of 214 combinatorial interpretations of the Catalan numbers, which will appear along with 68 additional problems in a monograph to be published in 2015. The Catalan numbers have the longest entry in the Online Encyclopedia of Integer Sequences (OEIS). Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Ways to Insert Parentheses in Multiplication of n Elements Theorem There are Cn−1 ways to parenthesize multiplication of n elements into binary multiplication. Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Ways to Insert Parentheses in Multiplication of n Elements Theorem There are Cn−1 ways to parenthesize multiplication of n elements into binary multiplication. Proof idea: After showing base case n = 1, assume that there are Ck−1 ways to parenthesize multiplication of k elements for 1 ≤ k ≤ n − 1. Start with the last multiplication to be performed. This divides x1 · · · xn into (x1 · · · xk )(xk+1 · · · xn ) Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Ways to Insert Parentheses in Multiplication of n Elements Example: n = 6 a(bcdef ) Parenthesize a in C0 = 1 way and bcdef in C4 = 14 ways. (ab)(cdef ) Parenthesize ab in C1 = 1 way and cdef in C3 = 5 ways. (abc)(def ) Parenthesize abc in C2 = 2 ways and def in C2 = 2 ways. Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Ways to Insert Parentheses in Multiplication of n Elements (abcd)(ef ) Parenthesize abcd in C3 = 5 ways and ef in C1 = 1 way. (abcde)f Parenthesize abcde in C4 = 14 ways and f in C0 = 1 way. Thus, we get 1 · 14 + 1 · 5 + 2 · 2 + 5 · 1 + 14 · 1 = 42 = C5 ways to parenthesize abcdef . Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Ways to Insert Parentheses in Multiplication of n Elements (abcd)(ef ) Parenthesize abcd in C3 = 5 ways and ef in C1 = 1 way. (abcde)f Parenthesize abcde in C4 = 14 ways and f in C0 = 1 way. Thus, we get 1 · 14 + 1 · 5 + 2 · 2 + 5 · 1 + 14 · 1 = 42 = C5 ways to parenthesize abcdef . In general, get n−2 P Ck Cn−2−k = Cn−1 ways to parenthesize k=0 x1 · · · xn by this process. Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Ways to Insert Parentheses in Multiplication of n Elements For n = 5, there are 14 ways. a(((bc)d)e) a(b(c(de))) a((bc)(de)) a((b(cd))e) a(b((cd)e)) (ab)((cd)e) (ab)(c(de)) ((ab)c)(de) (a(bc))(de) (((ab)c)d)e (a(b(cd)))e ((ab)(cd))e ((a(bc))d)e (a((bc)d))e Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Polygon Triangulations Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Polygon Triangulations Theorem There are Cn−2 ways to triangulate a convex n-gon. Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Polygon Triangulations Theorem There are Cn−2 ways to triangulate a convex n-gon. This is proven by removing an edge from the polygon to split it up into triangulations of polygons with fewer sides. Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Polygon Triangulations Theorem There are Cn−2 ways to triangulate a convex n-gon. This is proven by removing an edge from the polygon to split it up into triangulations of polygons with fewer sides. Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Balls into Boxes Theorem There are Cn ways to put n indistinguishable balls into n boxes labeled B1 , B2 , . . . , Bn such that the collection B1 , . . . , Bi has at most i balls, for any i ∈ {1, . . . , n}. Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Nervous Politician 2n votes cast in a political race. Politicians A and B each receive n votes. In how many orders can the votes be counted, so that A is never trailing B? Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Nervous Politician 2n votes cast in a political race. Politicians A and B each receive n votes. In how many orders can the votes be counted, so that A is never trailing B? These ways of counting the votes, subject to the above conditions, are called ballot sequences. Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Nervous Politician 2n votes cast in a political race. Politicians A and B each receive n votes. In how many orders can the votes be counted, so that A is never trailing B? These ways of counting the votes, subject to the above conditions, are called ballot sequences. Example When n = 3, there are C3 = 5 ways! AAABBB, AABABB, AABBAB, ABAABB, ABABAB Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Nervous Politician Theorem Given n, there are Cn ballot sequences. Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Nervous Politician Theorem Given n, there are Cn ballot sequences. To split a ballot sequence with 2(n + 1) votes up into smaller ballot sequences, Start with the A at the beginning. AAABABBABAABBBABAABB Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Nervous Politician Theorem Given n, there are Cn ballot sequences. To split a ballot sequence with 2(n + 1) votes up into smaller ballot sequences, Start with the A at the beginning. Find the place where the first tie occurs. AAABABBABAABBBABAABB Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Nervous Politician Theorem Given n, there are Cn ballot sequences. To split a ballot sequence with 2(n + 1) votes up into smaller ballot sequences, Start with the A at the beginning. Find the place where the first tie occurs. Take out this A and B, and divide it into smaller ballot sequences. AAABABBABAABBBABAABB Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Nervous Politician Theorem Given n, there are Cn ballot sequences. To split a ballot sequence with 2(n + 1) votes up into smaller ballot sequences, Start with the A at the beginning. Find the place where the first tie occurs. Take out this A and B, and divide it into smaller ballot sequences. AABABBABAABB Mike Joseph ABAABB Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Nervous Politician Given that there are Ck ballot sequences with 2k votes, for 0 ≤ k ≤ n, it follows that the number of ballot sequences with 2(n + 1) votes is n X Ck Cn−k = Cn+1 . k=0 Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Dyck Paths A Dyck path of length 2n is a lattice path from (0, 0) to (2n, 0) with steps (1, 1) and (1, −1) that never falls below the x-axis. Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Dyck Paths A Dyck path of length 2n is a lattice path from (0, 0) to (2n, 0) with steps (1, 1) and (1, −1) that never falls below the x-axis. Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Dyck Paths A Dyck path of length 2n is a lattice path from (0, 0) to (2n, 0) with steps (1, 1) and (1, −1) that never falls below the x-axis. The number of Dyck paths of length 2n is Cn , as they are in bijection with ballot sequences with 2n votes. Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Binary Trees A binary tree is a tree in which each node has at most two children. Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Binary Trees A binary tree is a tree in which each node has at most two children. Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Binary Trees A binary tree is a tree in which each node has at most two children. The number of binary trees with n nodes is Cn . Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References 312-Avoiding Permutations A permutation a1 a2 · · · an of 1, 2, . . . , n is called 312-avoiding if there does not exist i < j < k for which aj < ak < ai . Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References 312-Avoiding Permutations A permutation a1 a2 · · · an of 1, 2, . . . , n is called 312-avoiding if there does not exist i < j < k for which aj < ak < ai . Otherwise, the permutation contains the pattern 312 if there exist i < j < k for which aj < ak < ai . Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References 312-Avoiding Permutations A permutation a1 a2 · · · an of 1, 2, . . . , n is called 312-avoiding if there does not exist i < j < k for which aj < ak < ai . Otherwise, the permutation contains the pattern 312 if there exist i < j < k for which aj < ak < ai . Example 2167435 contains the pattern 312. Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References 312-Avoiding Permutations A permutation a1 a2 · · · an of 1, 2, . . . , n is called 312-avoiding if there does not exist i < j < k for which aj < ak < ai . Otherwise, the permutation contains the pattern 312 if there exist i < j < k for which aj < ak < ai . Example 2167435 contains the pattern 312. Example 2431657 is 312-avoiding. Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References 312-Avoiding Permutations A permutation a1 a2 · · · an of 1, 2, . . . , n is called 312-avoiding if there does not exist i < j < k for which aj < ak < ai . Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References 312-Avoiding Permutations A permutation a1 a2 · · · an of 1, 2, . . . , n is called 312-avoiding if there does not exist i < j < k for which aj < ak < ai . For n = 3, 123 132 213 Mike Joseph 231 Catalan Numbers 321 Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References 312-Avoiding Permutations A permutation a1 a2 · · · an of 1, 2, . . . , n is called 312-avoiding if there does not exist i < j < k for which aj < ak < ai . For n = 3, 123 132 213 231 1324 2431 1342 3214 1432 3241 321 For n = 4, 1234 2314 1243 2341 Mike Joseph Catalan Numbers 2134 3421 2143 4321 Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References 312-Avoiding Permutations Theorem There are Cn 312-avoiding permutations in Sn . Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References 312-Avoiding Permutations Theorem There are Cn 312-avoiding permutations in Sn . To show the recurrence relation, split a 312-avoiding permutation up by removing the 1. Example 234651987 → 23465 987 Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References 312-Avoiding Permutations 234651987 → 23465 987 → (12354, 321) To go the other way, (543261, 12) → 654372 Mike Joseph 89 → 654372189 Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References 321-Avoiding Permutations A permutation a1 a2 · · · an of 1, 2, . . . , n is called 321-avoiding if there does not exist i < j < k for which ak < aj < ai . Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References 321-Avoiding Permutations A permutation a1 a2 · · · an of 1, 2, . . . , n is called 321-avoiding if there does not exist i < j < k for which ak < aj < ai . Theorem There are Cn 321-avoiding permutations in Sn . Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Non-crossing Matchings Given 2n points in a row, pair them with non-crossing arcs above the points. The number of ways to do this is Cn . Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Non-crossing Matchings Bijection between non-crossing matchings and ballot sequences, by writing L for each left endpoint and R for each right endpoint. → Mike Joseph LRLLRR Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Non-crossing Matchings Inverse bijection: Scan from right to left. Match every L with the leftmost available R to the right of it. LRLLRR → Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Non-nesting Matchings Given 2n points in a row, pair them with non-nesting arcs above the points. The number of ways to do this is also Cn . Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Non-nesting Matchings Bijection between non-nesting matchings and ballot sequences, by writing L for each left endpoint and R for each right endpoint. → Mike Joseph LRLLRR Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Non-nesting Matchings Inverse bijection: Scan from right to left. Match every L with the rightmost available R to the right of it. LRLLRR → Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References There are 214 (and counting) known combinatorial interpretations of Catalan numbers. Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References There are 214 (and counting) known combinatorial interpretations of Catalan numbers. So that means we have them! 214 2 Mike Joseph total bijections between Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Parenthesizations of multiplication ←→ of n + 1 elements (ab)(cd) ((ab)c)d a(b(cd)) (a(bc))d a((bc)d) Mike Joseph ←→ ←→ ←→ ←→ ←→ Ballot sequences with 2n votes AABBAB AAABBB ABABAB AABABB ABAABB Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Parenthesizations of multiplication ←→ of n + 1 elements ((a · b) · (c · d)) (((a · b) · c) · d) (a · (b · (c · d))) ((a · (b · c)) · d) (a · ((b · c) · d)) Mike Joseph ←→ ←→ ←→ ←→ ←→ Ballot sequences with 2n votes AABBAB AAABBB ABABAB AABABB ABAABB Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Parenthesizations of multiplication ←→ of n + 1 elements ((a·b)·(c·d)) (((a·b)·c)·d) (a·(b·(c·d))) ((a·(b·c))·d) (a·((b·c)·d)) Mike Joseph ←→ ←→ ←→ ←→ ←→ Ballot sequences with 2n votes AABBAB AAABBB ABABAB AABABB ABAABB Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Connections with Number Theory Let b(n) be the number of 1s in the binary expansion of n. The exponent of the largest power of 2 dividing Cn is b(n + 1) − 1. Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Connections with Number Theory Let b(n) be the number of 1s in the binary expansion of n. The exponent of the largest power of 2 dividing Cn is b(n + 1) − 1. Proving this uses Kummer’s theorem. (Given n ≥ m ≥ 0 and prime p, the maximum integer k such that pk divides mn is the number of carries when m is added to n − m in base p.) Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Connections with Number Theory Let b(n) be the number of 1s in the binary expansion of n. The exponent of the largest power of 2 dividing Cn is b(n + 1) − 1. Proving this uses Kummer’s theorem. (Given n ≥ m ≥ 0 and prime p, the maximum integer k such that pk divides mn is the number of carries when m is added to n − m in base p.) This shows that Cn is odd if and only if n + 1 is a power of 2. Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Connections with Number Theory If f (n) is the number of positive integers k ≤ n that can be expressed as a sum of three squares of nonnegative integers. Then f (n) 5 = . n→∞ n 6 lim Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Connections with Number Theory If f (n) is the number of positive integers k ≤ n that can be expressed as a sum of three squares of nonnegative integers. Then f (n) 5 = . n→∞ n 6 lim Legendre’s theorem: A positive integer is the sum of three squares if and only if it is not of the form 4a (8b + 7) for nonnegative integers a, b. Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Connections with Number Theory If g(n) is the number of positive integers k ≤ n such that Ck is the sum of three squares of nonnegative integers. Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Connections with Number Theory If g(n) is the number of positive integers k ≤ n such that Ck is the sum of three squares of nonnegative integers. Then g(n) 7 = . n→∞ n 8 lim Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Connections with Number Theory 1, 1, 2, 5, 14, 42, 132, 429, . . . C0 |C1 C1 |C2 C4 |C5 1|1 1|2 14|42 For what n do we have Cn |Cn+1 ? Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Connections with Number Theory Cn |Cn+1 only for n = 0, 1, 4 Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Connections with Number Theory Cn |Cn+1 only for n = 0, 1, 4 Recall that Cn+1 = integer. 4n+2 C . n+2 n So Cn |Cn+1 only when Mike Joseph Catalan Numbers 4n+2 n+2 is an Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Connections with Number Theory Since 4(n + 2) > 4n + 2, we can only have 4n + 2 = k(n + 2) for k = 1, 2, 3. Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References Connections with Number Theory Since 4(n + 2) > 4n + 2, we can only have 4n + 2 = k(n + 2) for k = 1, 2, 3. 4n + 2 = 1(n + 2) ⇒ n = 0 4n + 2 = 2(n + 2) ⇒ n = 1 4n + 2 = 3(n + 2) ⇒ n = 4 Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References A Catalan Series ∞ X 1 1 1 1 1 1 = 1+1+ + + + + +··· = Cn 2 5 14 42 132 n=0 Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References A Catalan Series √ ∞ X 1 1 1 1 1 1 4 3π = 1+1+ + + + + +··· = 2+ Cn 2 5 14 42 132 27 n=0 Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References A Catalan Series Consequence of √ ∞ −1 1 √ X x 2x + 16 24 x sin xn 2 + = 2 5/2 C (4 − x) (4 − x) n n=0 Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References A Catalan Series Consequence of √ ∞ −1 1 √ X x 2x + 16 24 x sin xn 2 + = 2 5/2 C (4 − x) (4 − x) n n=0 ∞ X 4 − 3n n=0 Mike Joseph Cn =2 Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References OEIS One place to learn more about Catalan numbers is by looking them up in the Online Encyclopedia of Integer Sequences (OEIS). http://oeis.org Mike Joseph Catalan Numbers Introduction History and Importance Combinatorial Interpretations Connections with Number Theory and Analysis References References Sloane, N. J. A. "A000108 - OEIS." A000108 - OEIS. Web. 2 Dec. 2014. <http://oeis.org/A000108>. Stanley, Richard P. Enumerative Combinatorics. Second ed. Vol. 1. New York: Cambridge UP, 2012. Print. Stanley, Richard P. "Some Catalan Musings." Institute for Mathematics and its Applications. 10 Nov. 2014. Web. <http://www.ima.umn.edu/videos/?id=2833>. Mike Joseph Catalan Numbers