Lattice Paths and Their Generalizations

Transcription

1 and Their Generalizations SeungKyung Park Yonsei University August 9, 2012

2 Lattice paths Google search : lattice paths About 11,700,000 results (0.18 seconds)

3 Lattice paths Google search : lattice paths About 11,700,000 results (0.18 seconds) Google search : lattice paths genelization About 4,610,000 results (0.29 seconds)

4 Lattice paths Google search : lattice paths About 11,700,000 results (0.18 seconds) Google search : lattice paths genelization About 4,610,000 results (0.29 seconds) Google search : generalized lattice paths About 4,590,000 results (0.27 seconds)

5 Lattice paths Google search : lattice paths About 11,700,000 results (0.18 seconds) Google search : lattice paths genelization About 4,610,000 results (0.29 seconds) Google search : generalized lattice paths About 4,590,000 results (0.27 seconds) Google search : lattice paths About 1,280 results (0.25 seconds)

6 Lattice paths Google search : lattice paths About 11,700,000 results (0.18 seconds) Google search : lattice paths genelization About 4,610,000 results (0.29 seconds) Google search : generalized lattice paths About 4,590,000 results (0.27 seconds) Google search : lattice paths About 1,280 results (0.25 seconds) Google search : ***** About 41,800,000 results(0.25 seconds)

7 The Ballot Problem Suppose that candidate A gets m votes and candidate B receives n votes in an election. What is the probability that the ballots of A is always ahead of B throughout the counting? Transform the situation to lattice paths

8 The Ballot Problem Suppose that candidate A gets m votes and candidate B receives n votes in an election. What is the probability that the ballots of A is always ahead of B throughout the counting? Transform the situation to lattice paths a vote for A : (0,1),

9 The Ballot Problem Suppose that candidate A gets m votes and candidate B receives n votes in an election. What is the probability that the ballots of A is always ahead of B throughout the counting? Transform the situation to lattice paths a vote for A : (0,1), a vote for B : (1,0),

10 The Ballot Problem Suppose that candidate A gets m votes and candidate B receives n votes in an election. What is the probability that the ballots of A is always ahead of B throughout the counting? Transform the situation to lattice paths a vote for A : (0,1), a vote for B : (1,0), the countings = the number of paths from (0, 0) to (m, n).

11 The Ballot Problem Suppose that candidate A gets m votes and candidate B receives n votes in an election. What is the probability that the ballots of A is always ahead of B throughout the counting? Transform the situation to lattice paths a vote for A : (0,1), a vote for B : (1,0), the countings = the number of paths from (0, 0) to (m, n). Since m n 0, if M is the number of paths from (1, 0) to (m, n) that never touch the line x = y, then the probability is M ( m+n m ).

12 The Ballot Problem Suppose that candidate A gets m votes and candidate B receives n votes in an election. What is the probability that the ballots of A is always ahead of B throughout the counting? Transform the situation to lattice paths a vote for A : (0,1), a vote for B : (1,0), the countings = the number of paths from (0, 0) to (m, n). Since m n 0, if M is the number of paths from (1, 0) to (m, n) that never touch the line x = y, then the probability is M ( m+n m ). How to count M?

13 The Reflection Principle Andre s solution(1887): the reflection principle If a path touches or goes over the line x = y, find the first point that crosses the line.

14 The Reflection Principle Andre s solution(1887): the reflection principle If a path touches or goes over the line x = y, find the first point that crosses the line. Then reflect this portion to the line x = y.

15 The Reflection Principle Andre s solution(1887): the reflection principle If a path touches or goes over the line x = y, find the first point that crosses the line. Then reflect this portion to the line x = y. There is a bijection between paths that start at (1, 0) and touch the line and paths that start at (0, 1).

16 The Reflection Principle Andre s solution(1887): the reflection principle If a path touches or goes over the line x = y, find the first point that crosses the line. Then reflect this portion to the line x = y. There is a bijection between paths that start at (1, 0) and touch the line and paths that start at (0, 1). Thus ( ) ( ) m 1 + n m + n 1 M = m 1 m = m n m + n ( ) m + n m

17 The Reflection Principle Andre s solution(1887): the reflection principle If a path touches or goes over the line x = y, find the first point that crosses the line. Then reflect this portion to the line x = y. There is a bijection between paths that start at (1, 0) and touch the line and paths that start at (0, 1). Thus ( ) m 1 + n M = m 1 ( ) m + n 1 m Therefore the probability is m n m + n. = m n m + n ( ) m + n m

18 The Ballot Problem

19 The Catalan Number ( ) 2n + 1 = 1 n + 1 n + 1 ( ) 2n = C n, n If m = n + 1, then M = 1 2n + 1 the Catalan number = the number of paths from (0, 0) to (n, n) that never go above the line x = y. The word lattice paths was used first by MacMahon(1909). He showed these are related to permutations, combinations, partitions, and certain probabilities.

20 Generalizing the Ballot Problem There are many directions to generalizing the problem

21 Generalizing the Ballot Problem There are many directions to generalizing the problem The case that has multicandidates

22 Generalizing the Ballot Problem There are many directions to generalizing the problem The case that has multicandidates Replacing the line x = y by x = ky

23 Generalizing the Ballot Problem There are many directions to generalizing the problem The case that has multicandidates Replacing the line x = y by x = ky Lattice paths in Weyl groups

24 Generalizing the Ballot Problem There are many directions to generalizing the problem The case that has multicandidates Replacing the line x = y by x = ky Lattice paths in Weyl groups Super Ballot numbers (k + 2r)! (2n + k 1)! (k 1)!r! n!(n + k + r)!

25 The Methodology The Methods Solutions = an explicit formula, a recurrence, a generating function, and an asymtotic formula The Methods Bijections like the reflection principle

26 The Methodology The Methods Solutions = an explicit formula, a recurrence, a generating function, and an asymtotic formula The Methods Bijections like the reflection principle Method of images

27 The Methodology The Methods Solutions = an explicit formula, a recurrence, a generating function, and an asymtotic formula The Methods Bijections like the reflection principle Method of images Cycle lemma

28 The Methodology The Methods Solutions = an explicit formula, a recurrence, a generating function, and an asymtotic formula The Methods Bijections like the reflection principle Method of images Cycle lemma Generating functions(including monoid theory)

29 The Methodology The Methods Solutions = an explicit formula, a recurrence, a generating function, and an asymtotic formula The Methods Bijections like the reflection principle Method of images Cycle lemma Generating functions(including monoid theory) Lagrange Inversion Formula f(x) = xg(f(x)) [x n ]f(x) k = k n [tn k ]G(t) n

30 The Methodology The Methods Solutions = an explicit formula, a recurrence, a generating function, and an asymtotic formula The Methods Bijections like the reflection principle Method of images Cycle lemma Generating functions(including monoid theory) Lagrange Inversion Formula f(x) = xg(f(x)) [x n ]f(x) k = k n [tn k ]G(t) n Probabilistic method

31 The Methodology The Methods Solutions = an explicit formula, a recurrence, a generating function, and an asymtotic formula The Methods Bijections like the reflection principle Method of images Cycle lemma Generating functions(including monoid theory) Lagrange Inversion Formula f(x) = xg(f(x)) [x n ]f(x) k = k n [tn k ]G(t) n Probabilistic method Kernel method(k(x, y)f (x, y) = A(x, y)g(x, y) + B(x, y))

32 The Methodology Applications Pattern avoiding permutations

33 The Methodology Applications Pattern avoiding permutations Riordan arrays(matrices)

34 The Methodology Applications Pattern avoiding permutations Riordan arrays(matrices) Non-intersecting paths(determinants)

35 The Methodology Applications Pattern avoiding permutations Riordan arrays(matrices) Non-intersecting paths(determinants) Young tableaux

36 The Methodology Applications Pattern avoiding permutations Riordan arrays(matrices) Non-intersecting paths(determinants) Young tableaux Rooted trees

37 The Methodology Applications Pattern avoiding permutations Riordan arrays(matrices) Non-intersecting paths(determinants) Young tableaux Rooted trees Many others CATALAN ADDENDUM by Richard P. Stanley version of 13 July 2012 (The number is 136 and increasing!)

38 Generalizations Dimensions The lattice paths in d-dimensional Cartesian spaces d = 1 : a line

39 Generalizations Dimensions The lattice paths in d-dimensional Cartesian spaces d = 1 : a line d = 2 : ordinary plane paths

40 Generalizations Dimensions The lattice paths in d-dimensional Cartesian spaces d = 1 : a line d = 2 : ordinary plane paths d = 3 : Krewaras (1965), Gessel(1986), Sulanke(2005)

41 Generalizations Dimensions The lattice paths in d-dimensional Cartesian spaces d = 1 : a line d = 2 : ordinary plane paths d = 3 : Krewaras (1965), Gessel(1986), Sulanke(2005) d = n : Zeilberger(1983), Watanabe & Mohanty(1987)

42 Generalizations The Step sets Consider lattice paths with different step sets S: Two step sets : S = {(1, 1), (1, 1)}(or S = {(1, 0), (0, 1)}) Dyck paths are lattice paths with the step set S from (0, 0) to (2n, 0) that do not go below the x-axis.

43 Generalizations The Step sets Consider lattice paths with different step sets S: Two step sets : S = {(1, 1), (1, 1)}(or S = {(1, 0), (0, 1)}) Dyck paths are lattice paths with the step set S from (0, 0) to (2n, 0) that do not go below the x-axis. Three step sets : S = {(1, 1), (1, 1), (1, 0)} Motzkin paths are lattice paths with the step set S from (0, 0) to (n, 0) that do not go below the x-axis.

44 Generalizations The Step sets Consider lattice paths with different step sets S: Two step sets : S = {(1, 1), (1, 1)}(or S = {(1, 0), (0, 1)}) Dyck paths are lattice paths with the step set S from (0, 0) to (2n, 0) that do not go below the x-axis. Three step sets : S = {(1, 1), (1, 1), (1, 0)} Motzkin paths are lattice paths with the step set S from (0, 0) to (n, 0) that do not go below the x-axis. S = {(2, 0), (0, 2), (1, 1)} Paths with this step set S that stay above the line x = y is also M n, the number of Motzkin paths.

45 Generalizations The Step sets Colored Motzkin paths

46 Generalizations The Step sets Colored Motzkin paths S = {(1, 1), (1, 1), (w, 0)} - Barcucci et al(2001, 2002)

47 Generalizations The Step sets Colored Motzkin paths S = {(1, 1), (1, 1), (w, 0)} - Barcucci et al(2001, 2002) A Schröder path is a lattice path with step set S = {(1, 0), (0, 1), (1, 1)} that stays weakly above the line x = y.

48 Generalizations The Step sets Colored Motzkin paths S = {(1, 1), (1, 1), (w, 0)} - Barcucci et al(2001, 2002) A Schröder path is a lattice path with step set S = {(1, 0), (0, 1), (1, 1)} that stays weakly above the line x = y. The (central) Delannoy numbers is the number of paths with S = {(1, 0), (0, 1), (1, 1)} that end at (n, n).

49 Generalizations The Step sets Colored Motzkin paths S = {(1, 1), (1, 1), (w, 0)} - Barcucci et al(2001, 2002) A Schröder path is a lattice path with step set S = {(1, 0), (0, 1), (1, 1)} that stays weakly above the line x = y. The (central) Delannoy numbers is the number of paths with S = {(1, 0), (0, 1), (1, 1)} that end at (n, n). S = {(m, 1), (m, 1), (k, 0)} - Labelle(1993)

50 Generalizations The Step sets More than three steps Paths with S = {(1, 0), (0, 1), (i, 1) i = 1, 2,, k} that stay under the line x = ry - Mohanty(1968) Paths with S = {(2, 1), (2, 1), (1, 2), (1, 2)} that stay above the x-axis - Labelle and Yeh(1989)(and the case of S = {(r, s), (r, s), (s, r), (s, r)}) S = {(m 1, 0), (m 2, 1), (m 2, 1), (m 3, 2), (m 3, 2)} - Labelle(1993)

51 Generalizations Recent results Combinatorially meaningful recent results Coker(2003) The number d n of lattice paths with the step set S = {(k, 0), (0, k)} that never go above the line x = y is n ( )( ) 1 n n d n = 4 n k. n k k 1 k=1

52 Generalizations Recent results Combinatorially meaningful recent results Rukavicka(2010) Let h n = {(t 1, l 1 ),, (t m, l m )}, where ( t i l i = n with ) 1 n + k l i l j if i j. Then there are paths n + 1 n, t 1,, t m that stay below the line x = y.(t i horizontal steps of the size l i with the unit vertical steps)

53 Generalizations Recent results Combinatorially meaningful recent results Ma and Yeh(2009) Generalized Chung-Feller Theorem(analytical method)

54 Generalizations Recent results Combinatorially meaningful recent results Ma and Yeh(2009) Generalized Chung-Feller Theorem(analytical method) Huq(2009) Generalized Chung-Feller Theorem(with the step set S = {(1, 1), (1, r)})

55 The Chung-Feller Theorem The Chung-Feller Theorem The Theorem(1949) ( ) 1 2n The number of lattice paths from (0, 0) to (2n, 0) is n + 1 n and independent from the number of flaws which are upsteps below the x-axis. Proved first by MacMahom in 1909 and later by Chung and Feller using generating functions. There are many bijective proofs.

56 The Narayana number The Narayana number The Narayana number N(n, k) is the number of Dyck paths that have k peaks. N(n, k) = 1 ( )( ) ( )( ) n n 1 1 n n 1 = k k 1 k 1 n k + 1 k k 1 ( )( ) 1 n + 1 n 1 = = 1 ( )( ) n n 1 n + 1 k k 1 k 1 k k 2 ( )( ) 1 n n 1 = = 1 ( )( ) n n. n k k 1 k n k k 1 The Catalan number : C n = n k=1 ( )( ) 1 n n n k k 1

57 The Catalan number Generalized Narayana and Catalan numbers Generalized Narayana numbers N r (n, k) = 1 ( )( ) rn n 1 = k k 1 k 1 ( )( ) 1 rn + 1 n 1 = rn + 1 k k 1 ( )( ) 1 rn n 1 = = 1 n k k 1 k n 1 rn k + 1 ( rn k ( )( ) n rn. k k 1 )( n 1 k 1 ) Generalized Catalan numbers ( 1 C r (n) = (r+1)n+1 (r+1)n+1 n ) = 1 rn+1 ( (r+1)n n ) = 1 n ( (r+1)n ) n 1

58 Other Paths Motzkin, Schröder, and Riordan Paths Motzkin paths : Paths with S = {(1, 1), (1, 1), (1, 0)} that never go below the x-axis

59 Other Paths Motzkin, Schröder, and Riordan Paths Motzkin paths : Paths with S = {(1, 1), (1, 1), (1, 0)} that never go below the x-axis Schröder paths : Paths with S = {(1, 1), (1, 1), (2, 0)} that never go below the x-axis

60 Other Paths Motzkin, Schröder, and Riordan Paths Motzkin paths : Paths with S = {(1, 1), (1, 1), (1, 0)} that never go below the x-axis Schröder paths : Paths with S = {(1, 1), (1, 1), (2, 0)} that never go below the x-axis Riordan paths : Motzkin paths with no flat steps on the x-axis

61 Other Paths Motzkin, Schröder, and Riordan Paths Motzkin paths : Paths with S = {(1, 1), (1, 1), (1, 0)} that never go below the x-axis Schröder paths : Paths with S = {(1, 1), (1, 1), (2, 0)} that never go below the x-axis Riordan paths : Motzkin paths with no flat steps on the x-axis Small Schröder paths : Schroder paths with no flat steps on the x-axis

62 Motzkin, Schröder, and Riordan Paths Enumeration of Motzkin, Schröder, and Riordan Paths M(n, k)= the number of Motzkin paths with k upsteps, k downsteps, and n 2k flat steps from (0, 0) to (n, 0) M(n, k) = k+1( 1 n 2k ) 2k)( k

63 Motzkin, Schröder, and Riordan Paths Enumeration of Motzkin, Schröder, and Riordan Paths M(n, k)= the number of Motzkin paths with k upsteps, k downsteps, and n 2k flat steps from (0, 0) to (n, 0) M(n, k) = k+1( 1 n 2k ) 2k)( k R(n, k) = the number of Schröder paths with k upsteps, k downsteps, and n k flat steps from (0, 0) to (n, 0) R(n, k) = 1 ( n+k )( 2k ) k+1 2k k and M(n + k, k) = R(n, k)

64 Motzkin, Schröder, and Riordan Paths Enumeration of Motzkin, Schröder, and Riordan Paths M(n, k)= the number of Motzkin paths with k upsteps, k downsteps, and n 2k flat steps from (0, 0) to (n, 0) M(n, k) = k+1( 1 n 2k ) 2k)( k R(n, k) = the number of Schröder paths with k upsteps, k downsteps, and n k flat steps from (0, 0) to (n, 0) R(n, k) = 1 ( n+k )( 2k ) k+1 2k k and M(n + k, k) = R(n, k) J(n, k) = the number of Riordan paths J(n, k) = 1 ( n k 1 )( n ) k k 1 k 1

65 Motzkin, Schröder, and Riordan Paths Enumeration of Motzkin, Schröder, and Riordan Paths M(n, k)= the number of Motzkin paths with k upsteps, k downsteps, and n 2k flat steps from (0, 0) to (n, 0) M(n, k) = k+1( 1 n 2k ) 2k)( k R(n, k) = the number of Schröder paths with k upsteps, k downsteps, and n k flat steps from (0, 0) to (n, 0) R(n, k) = 1 ( n+k )( 2k ) k+1 2k k and M(n + k, k) = R(n, k) J(n, k) = the number of Riordan paths J(n, k) = 1 ( n k 1 )( n ) k k 1 k 1 S(n, k) = the number of small Schröder paths S(n, k) = 1 ( n 1 )( n+k k k 1 k 1) and J(n + k, k) = S(n, k)

66 Other Paths Gessel Walks Consider lattice paths with the step set S = (1, 1), (1, 0), (1, 0), (1, 1) from (0, 0) to (0, 0).

67 Other Paths Gessel Walks Gessel Conjecture(2001) The number of lattice paths with the step set S = (1, 0), ( 1, 0), (1, 1), ( 1, 1) from (0, 0) to (0, 0) with 2n steps is G(n) = 16 n (5/6) n(1/2) n (2) n (5/3) n, where (a) n = a(a + 1) (a + n 1). Proved by Zeilberger et al(2008) using computers

68 Other Paths Gessel Walks Gessel Conjecture(2001) The number of lattice paths with the step set S = (1, 0), ( 1, 0), (1, 1), ( 1, 1) from (0, 0) to (0, 0) with 2n steps is G(n) = 16 n (5/6) n(1/2) n (2) n (5/3) n, where (a) n = a(a + 1) (a + n 1). Proved by Zeilberger et al(2008) using computers If k is the number of occurences of steps (1, 1) and ( 1, 1), then only the cases of k = 1, 2, 3 have been proved combinatorially.

69 Current Research Generalizations S = {(r, r), (s, s) r, s P}

70 Current Research Generalizations S = {(r, r), (s, s) r, s P} S = {(a, r), (b, s) a, b, r, s P}

71 Current Research Generalizations S = {(r, r), (s, s) r, s P} S = {(a, r), (b, s) a, b, r, s P} How about other lattice paths with the above step sets.