Some Catalan Musings p. 1. Some Catalan Musings. Richard P. Stanley

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1 Some Catalan Musings p. 1 Some Catalan Musings Richard P. Stanley

2 Some Catalan Musings p. 2 An OEIS entry A000108: 1,1,2,5,14,42,132,429,...

3 Some Catalan Musings p. 2 An OEIS entry A000108: 1,1,2,5,14,42,132,429,... COMMENTS.... This is probably the longest entry in OEIS, and rightly so.

4 Some Catalan Musings p. 2 An OEIS entry A000108: 1,1,2,5,14,42,132,429,... COMMENTS.... This is probably the longest entry in OEIS, and rightly so. C n = 1 n+1( 2n n), n 0 (Catalan number)

5 Some Catalan Musings p. 3 Catalan monograph R. Stanley, Catalan Numbers, Cambridge University Press, to appear.

6 Some Catalan Musings p. 3 Catalan monograph R. Stanley, Catalan Numbers, Cambridge University Press, to appear. Includes 214 combinatorial interpretations of C n and 66 additional problems.

7 An early version (1970 s) Some Catalan Musings p. 4

8 An early version (1970 s) Some Catalan Musings p. 4

9 Some Catalan Musings p. 5

10 Some Catalan Musings p. 6 How to sample? Compare D. E. Knuth, 3:16 Bible Texts Illuminated. Sample from Bible by choosing verse 3:16 from each chapter.

11 Some Catalan Musings p. 6 How to sample? Compare D. E. Knuth, 3:16 Bible Texts Illuminated. Sample from Bible by choosing verse 3:16 from each chapter. I will be less random.

12 Some Catalan Musings p. 7 History Sharabiin Myangat, also known as Minggatu, Ming antu ( ), and Jing An (c c. 1763): a Mongolian astronomer, mathematician, and topographic scientist who worked at the Qing court in China.

13 Some Catalan Musings p. 7 History Sharabiin Myangat, also known as Minggatu, Ming antu ( ), and Jing An (c c. 1763): a Mongolian astronomer, mathematician, and topographic scientist who worked at the Qing court in China. Typical result (1730 s): sin(2α) = 2sinα n=1 C n 1 4 n 1 sin2n+1 α

14 Some Catalan Musings p. 7 History Sharabiin Myangat, also known as Minggatu, Ming antu ( ), and Jing An (c c. 1763): a Mongolian astronomer, mathematician, and topographic scientist who worked at the Qing court in China. Typical result (1730 s): sin(2α) = 2sinα n=1 C n 1 4 n 1 sin2n+1 α No combinatorics, no further work in China.

15 Some Catalan Musings p. 8 More history, via Igor Pak Euler (1751): conjectured formula for number C n of triangulations of a convex (n+2)-gon

16 Some Catalan Musings p. 9 Completion of proof Goldbach and Segner ( ): helped Euler complete the proof, in pieces. Lamé (1838): first self-contained, complete proof.

17 Some Catalan Musings p. 10 Catalan Eugéne Charles Catalan (1838): wrote C n in the form and showed they counted (2n)! n!(n+1)! (nonassociative) bracketings (or parenthesizations) of a string of n+1 letters.

18 Some Catalan Musings p. 11 Why Catalan numbers? Riordan (1948): introduced the term Catalan number in Math Reviews.

19 Some Catalan Musings p. 11 Why Catalan numbers? Riordan (1948): introduced the term Catalan number in Math Reviews. Riordan (1964): used the term again in Math. Reviews.

20 Some Catalan Musings p. 11 Why Catalan numbers? Riordan (1948): introduced the term Catalan number in Math Reviews. Riordan (1964): used the term again in Math. Reviews. Riordan (1968): used the term in his book Combinatorial Identities. Finally caught on.

21 Some Catalan Musings p. 11 Why Catalan numbers? Riordan (1948): introduced the term Catalan number in Math Reviews. Riordan (1964): used the term again in Math. Reviews. Riordan (1968): used the term in his book Combinatorial Identities. Finally caught on. Gardner (1976): used the term in his Mathematical Games column in Scientific American. Real popularity began.

22 Some Catalan Musings p. 12 The primary recurrence C n+1 = n k=0 C k C n k, C 0 = 1

23 Some Catalan Musings p. 12 The primary recurrence C n+1 = n k=0 C k C n k, C 0 = 1 e

24 Some Catalan Musings p. 13 Transparent interpretations 3. Binary parenthesizations or bracketings of a string of n+1 letters (xx x)x x(xx x) (x xx)x x(x xx) xx xx

25 Some Catalan Musings p. 13 Transparent interpretations 3. Binary parenthesizations or bracketings of a string of n+1 letters (xx x)x x(xx x) (x xx)x x(x xx) xx xx ((x(xx))x)(x(xx)(xx))

26 Some Catalan Musings p. 13 Transparent interpretations 3. Binary parenthesizations or bracketings of a string of n+1 letters (xx x)x x(xx x) (x xx)x x(x xx) xx xx ((x(xx))x)(x(xx)(xx))

27 Some Catalan Musings p. 14 Binary trees 4. Binary trees with n vertices

28 Some Catalan Musings p. 14 Binary trees 4. Binary trees with n vertices

29 Some Catalan Musings p. 14 Binary trees 4. Binary trees with n vertices

30 Some Catalan Musings p. 15 Dyck paths 25. Dyck paths of length 2n, i.e., lattice paths from (0, 0) to (2n, 0) with steps (1, 1) and (1, 1), never falling below the x-axis

31 Some Catalan Musings p. 15 Dyck paths 25. Dyck paths of length 2n, i.e., lattice paths from (0, 0) to (2n, 0) with steps (1, 1) and (1, 1), never falling below the x-axis

32 Some Catalan Musings p. 15 Dyck paths 25. Dyck paths of length 2n, i.e., lattice paths from (0, 0) to (2n, 0) with steps (1, 1) and (1, 1), never falling below the x-axis

33 Some Catalan Musings p avoiding permutations 116. Permutations a 1 a 2 a n of 1,2,...,n for which there does not exist i < j < k and a j < a k < a i (called 312-avoiding) permutations)

34 Some Catalan Musings p avoiding permutations 116. Permutations a 1 a 2 a n of 1,2,...,n for which there does not exist i < j < k and a j < a k < a i (called 312-avoiding) permutations)

35 Some Catalan Musings p avoiding permutations 116. Permutations a 1 a 2 a n of 1,2,...,n for which there does not exist i < j < k and a j < a k < a i (called 312-avoiding) permutations)

36 Some Catalan Musings p. 17 Staircase tilings 205. Tilings of the staircase shape (n,n 1,...,1) with n rectangles

37 Some Catalan Musings p. 17 Staircase tilings 205. Tilings of the staircase shape (n,n 1,...,1) with n rectangles

38 Some Catalan Musings p. 17 Staircase tilings 205. Tilings of the staircase shape (n,n 1,...,1) with n rectangles

39 Some Catalan Musings p. 18 Less transparent interpretations Plane tree: subtrees of a vertex are linearly ordered 6. Plane trees with n + 1 vertices

40 Some Catalan Musings p. 19 The natural bijection a a b c a b c d b d e f g d e f g h e c h i j h i j i f j g

41 Some Catalan Musings p. 20 Noncrossing partitions 159. Noncrossing partitions of 1, 2,..., n, i.e., partitions π = {B 1,...,B k } Π n such that if a < b < c < d and a,c B i and b,d B j, then i = j

42 Bijection with plane trees Some Catalan Musings p. 21

43 Some Catalan Musings p. 21 Bijection with plane trees

44 Some Catalan Musings p. 21 Bijection with plane trees Children of nonleaf vertices: {1,5,6}, {2},{3,4},{7,9},{8},{10,11,12}

45 Some Catalan Musings p avoiding permutations 115. Permutations a 1 a 2 a n of 1,2,...,n with longest decreasing subsequence of length at most two (i.e., there does not exist i < j < k, a i > a j > a k ), called 321-avoiding permutations

46 Some Catalan Musings p. 23 Bijection with Dyck paths w =

47 Some Catalan Musings p. 23 Bijection with Dyck paths w =

48 Some Catalan Musings p. 23 Bijection with Dyck paths w =

49 Some Catalan Musings p. 24 Semiorders (finite) semiorder or unit interval order: a finite subset P of R with the partial order: x < P y x < R y 1 Equivalently, no induced (3 + 1) or (2 + 2)

50 Some Catalan Musings p. 25 Semiorders (cont.) 180. Nonisomorphic n-element posets with no induced subposet isomorphic to or 3 + 1

51 Some Catalan Musings p. 26 Semiorders and Dyck paths

52 Some Catalan Musings p. 26 Semiorders and Dyck paths x x x x x x x x x x

53 Some Catalan Musings p. 26 Semiorders and Dyck paths x x x x x x x x x x

54 Some Catalan Musings p. 27 Unexpected interpretations 92. n-tuples (a 1,a 2,...,a n ) of integers a i 2 such that in the sequence 1a 1 a 2 a n 1, each a i divides the sum of its two neighbors

55 Some Catalan Musings p. 27 Unexpected interpretations 92. n-tuples (a 1,a 2,...,a n ) of integers a i 2 such that in the sequence 1a 1 a 2 a n 1, each a i divides the sum of its two neighbors

56 Some Catalan Musings p. 27 Unexpected interpretations 92. n-tuples (a 1,a 2,...,a n ) of integers a i 2 such that in the sequence 1a 1 a 2 a n 1, each a i divides the sum of its two neighbors

57 Some Catalan Musings p. 27 Unexpected interpretations 92. n-tuples (a 1,a 2,...,a n ) of integers a i 2 such that in the sequence 1a 1 a 2 a n 1, each a i divides the sum of its two neighbors

58 Some Catalan Musings p. 27 Unexpected interpretations 92. n-tuples (a 1,a 2,...,a n ) of integers a i 2 such that in the sequence 1a 1 a 2 a n 1, each a i divides the sum of its two neighbors

59 Some Catalan Musings p. 27 Unexpected interpretations 92. n-tuples (a 1,a 2,...,a n ) of integers a i 2 such that in the sequence 1a 1 a 2 a n 1, each a i divides the sum of its two neighbors

60 Some Catalan Musings p. 27 Unexpected interpretations 92. n-tuples (a 1,a 2,...,a n ) of integers a i 2 such that in the sequence 1a 1 a 2 a n 1, each a i divides the sum of its two neighbors UDUUDDUD

61 Some Catalan Musings p. 28 Cores hook lengths of a partition λ p-core: a partition with no hook lengths equal to (equivalently, divisible by) p (p,q)-core: a partition that is simultaneously a p-core and q-core

62 Some Catalan Musings p. 29 (n,n + 1)-cores 112. Integer partitions that are both n-cores and (n+1)-cores

63 Some Catalan Musings p. 30 Constructing (5,6)-cores 6 5

64 Some Catalan Musings p. 30 Constructing (5,6)-cores

65 Some Catalan Musings p. 30 Constructing (5,6)-cores

66 Some Catalan Musings p. 30 Constructing (5,6)-cores

67 Some Catalan Musings p. 30 Constructing (5,6)-cores

68 Some Catalan Musings p. 31 (4,3,1,1,1,1) is a (5,6)-core

69 Some Catalan Musings p. 32 Inversions of permutations inversion of a 1 a 2 a n S n : (a i,a j ) such that i < j, a i > a j

70 Some Catalan Musings p. 32 Inversions of permutations inversion of a 1 a 2 a n S n : (a i,a j ) such that i < j, a i > a j 186. Sets S of n non-identity permutations in S n+1 such that every pair (i,j) with 1 i < j n is an inversion of exactly one permutation in S {1243, 2134, 3412},{1324, 2314, 4123},{2134, 3124, 4123} {1324, 1423, 2341},{1243, 1342, 2341}

71 Some Catalan Musings p. 32 Inversions of permutations inversion of a 1 a 2 a n S n : (a i,a j ) such that i < j, a i > a j 186. Sets S of n non-identity permutations in S n+1 such that every pair (i,j) with 1 i < j n is an inversion of exactly one permutation in S {1243, 2134, 3412},{1324, 2314, 4123},{2134, 3124, 4123} {1324, 1423, 2341},{1243, 1342, 2341} due to R. Dewji, I. Dimitrov, A. McCabe, M. Roth, D. Wehlau, J. Wilson

72 Some Catalan Musings p. 33 Tridiagonal matrices 207. n-tuples (a 1,...a n ) of positive integers such that the tridiagonal matrix a a a a n a n

73 Some Catalan Musings p. 34 Tridiagonal matrices (cont.) is positive definite with determinant one

74 Some Catalan Musings p. 35 A8. Algebraic interpretations (a) Number of two-sided ideals of the algebra of all (n 1) (n 1) upper triangular matrices over a field

75 Some Catalan Musings p. 35 A8. Algebraic interpretations (a) Number of two-sided ideals of the algebra of all (n 1) (n 1) upper triangular matrices over a field 0 * * * * * * * * * * * * * * * * * * * * *

76 Some Catalan Musings p. 36 Quasisymmetric functions Quasisymmetric function: a polynomial f Q[x 1,...,x n ] such that if i 1 < < i n then [x a 1 i 1 x a n i n ]f = [x a 1 1 xa n n ]f.

77 Some Catalan Musings p. 36 Quasisymmetric functions Quasisymmetric function: a polynomial f Q[x 1,...,x n ] such that if i 1 < < i n then [x a 1 i 1 x a n i n ]f = [x a 1 1 xa n n ]f. (k) Dimension (as a Q-vector space) of the ring Q[x 1,...,x n ]/Q n, where Q n denotes the ideal of Q[x 1,...,x n ] generated by all quasisymmetric functions in the variables x 1,...,x n with 0 constant term

78 Some Catalan Musings p. 36 Quasisymmetric functions Quasisymmetric function: a polynomial f Q[x 1,...,x n ] such that if i 1 < < i n then [x a 1 i 1 x a n i n ]f = [x a 1 1 xa n n ]f. (k) Dimension (as a Q-vector space) of the ring Q[x 1,...,x n ]/Q n, where Q n denotes the ideal of Q[x 1,...,x n ] generated by all quasisymmetric functions in the variables x 1,...,x n with 0 constant term Difficult proof by J.-C. Aval, F. Bergeron and N. Bergeron, 2004.

79 Some Catalan Musings p. 37 Diagonal harmonics (i) Let the symmetric group S n act on the polynomial ring A = C[x 1,...,x n,y 1,...,y n ] by w f(x 1,...,x n,y 1,...,y n ) = f(x w(1),...,x w(n),y w(1),...,y w(n) ) for all w S n. Let I be the ideal generated by all invariants of positive degree, i.e., I = f A : w f = f for all w S n, and f(0) = 0.

80 Some Catalan Musings p. 38 Diagonal harmonics (cont.) Then C n is the dimension of the subspace of A/I affording the sign representation, i.e., C n = dim{f A/I : w f = (sgnw)f for all w S n }.

81 Some Catalan Musings p. 38 Diagonal harmonics (cont.) Then C n is the dimension of the subspace of A/I affording the sign representation, i.e., C n = dim{f A/I : w f = (sgnw)f for all w S n }. Very deep proof by M. Haiman, 1994.

82 Some Catalan Musings p. 39 Number theory A61. Let b(n) denote the number of 1 s in the binary expansion of n. Using Kummer s theorem on binomial coefficients modulo a prime power, show that the exponent of the largest power of 2 dividing C n is equal to b(n+1) 1.

83 Some Catalan Musings p. 40 Sums of three squares Let f(n) denote the number of integers 1 k n such that k is the sum of three squares (of nonnegative integers). Well-known: lim n f(n) n = 5 6.

84 Sums of three squares Let f(n) denote the number of integers 1 k n such that k is the sum of three squares (of nonnegative integers). Well-known: lim n f(n) n = 5 6. Let g(n) denote the number of integers 1 k n such that C k is the sum of three squares. Then lim n g(n) n =??. Some Catalan Musings p. 40

85 Sums of three squares Let f(n) denote the number of integers 1 k n such that k is the sum of three squares (of nonnegative integers). Well-known: lim n f(n) n = 5 6. A63. Let g(n) denote the number of integers 1 k n such that C k is the sum of three squares. Then lim n g(n) n = 7 8. Some Catalan Musings p. 40

86 Some Catalan Musings p. 41 Analysis A65.(b) n 0 1 C n =??

87 Some Catalan Musings p. 41 Analysis A65.(b) n 0 1 = π C n 27.

88 Some Catalan Musings p. 42 Why? A65.(a) n 0 x n = 2(x+8) C n (4 x) + 24 ( xsin 1 1 ) 2 x. 2 (4 x) 5/2

89 Some Catalan Musings p. 42 Why? A65.(a) n 0 x n = 2(x+8) C n (4 x) + 24 ( xsin 1 1 ) 2 x. 2 (4 x) 5/2 Consequence of 2 ( sin 1 x ) 2 = 2 n 1 x 2n n 2( 2n n ).

90 Some Catalan Musings p. 42 Why? A65.(a) n 0 x n = 2(x+8) C n (4 x) + 24 ( xsin 1 1 ) 2 x. 2 (4 x) 5/2 Consequence of 2 ( sin 1 x ) 2 = 2 n 1 n 0 4 3n C n = 2. x 2n n 2( 2n n ).

91 An outlier Euler (1737): e = Convergents: 1,3, 19 7, ,... Some Catalan Musings p. 43

92 Some Catalan Musings p. 44 A curious generating function a n : numerator of the nth convergent a 1 = 1, a 2 = 3, a 3 = 19, a 4 = 193

93 Some Catalan Musings p. 44 A curious generating function a n : numerator of the nth convergent a 1 = 1, a 2 = 3, a 3 = 19, a 4 = n 1 a n x n n! = exp m 0 C m x m+1

94 The last slide Some Catalan Musings p. 45

95 The last slide Some Catalan Musings p. 45