Enumerative and Algebraic Combinatorics of OEIS A071356

Transcription

1 Enumerative and Algebraic Combinatorics of OEIS A Chetak Hossain Department of Matematics North Carolina State University July 9, 2018 Chetak Hossain (NCSU) Combinatorics of OEIS A July 9, / 30

2 Integer Sequences The Catalan numbers (A000108): 1, 1, 2, 5, 14, 42,... c n x n = 1 + x + 2x 2 + 5x x 4 + = 1 1 4x 2x n=0 The sequence A is: 1, 1, 2, 6, 20, 72,... a n x n = 1 + x + 2x 2 + 6x x 4 + = 2x x 4x 2 4x n=0 Chetak Hossain (NCSU) Combinatorics of OEIS A July 9, / 30

3 Inversion Sequences Definition The set of inversion sequences is I n = {(e 1,, e n ) N n e i i 1} A bijection between inversion sequences and permutations is to use the Lehmer code where we define and then reverse the sequence. e i = {j j < i and π j < π i } Chetak Hossain (NCSU) Combinatorics of OEIS A July 9, / 30

4 Weakly Increasing Inversion Sequences Definition Let C n = {(e 1,..., e n ) I n e 1... e n } be the set of weakly increasing inversion sequences. Chetak Hossain (NCSU) Combinatorics of OEIS A July 9, / 30

5 Weakly Increasing Inversion Sequences Definition Let C n = {(e 1,..., e n ) I n e 1... e n } be the set of weakly increasing inversion sequences. Theorem The reverse Lehmer code bijection restricts to a bijection between elements of S n (132) and weakly increasing inversion sequences. Corollary C n = c n Chetak Hossain (NCSU) Combinatorics of OEIS A July 9, / 30

6 Pattern Avoiding Inversion Sequences Definition We recall from [Martinez-Savage 2017] that elements of I n (e i > e j e k ) take the following form: e 1 e t > e t+1 > > e n for some t such that 1 < t n. Let t be called the peak of such an inversion sequence. Question (Martinez-Savage 2017) Is I n (e i > e j e k ) counted by OEIS A071356? Chetak Hossain (NCSU) Combinatorics of OEIS A July 9, / 30

7 Theorem (H. 2018) The generating function of I R n (e i e j < e k ) is 2x x 4x 2. 4x Chetak Hossain (NCSU) Combinatorics of OEIS A July 9, / 30

8 Theorem (H. 2018) The generating function of I R n (e i e j < e k ) is 2x x 4x 2. 4x Proof. Chetak Hossain (NCSU) Combinatorics of OEIS A July 9, / 30

9 Theorem (H. 2018) The generating function of I R n (e i e j < e k ) is 2x x 4x 2. 4x Proof. Given an inversion sequence of length n and peak t, there are two ways to break the sequence into two pieces so that the left piece is weakly increasing and the right piece is strictly decreasing. Chetak Hossain (NCSU) Combinatorics of OEIS A July 9, / 30

10 Theorem (H. 2018) The generating function of I R n (e i e j < e k ) is 2x x 4x 2. 4x Proof. Given an inversion sequence of length n and peak t, there are two ways to break the sequence into two pieces so that the left piece is weakly increasing and the right piece is strictly decreasing. e 1 e t 1 e t > e t+1 > > e n Chetak Hossain (NCSU) Combinatorics of OEIS A July 9, / 30

11 Theorem (H. 2018) The generating function of I R n (e i e j < e k ) is 2x x 4x 2. 4x Proof. Given an inversion sequence of length n and peak t, there are two ways to break the sequence into two pieces so that the left piece is weakly increasing and the right piece is strictly decreasing. e 1 e t 1 e t > e t+1 > > e n e 1 e t e t+1 > > e n Chetak Hossain (NCSU) Combinatorics of OEIS A July 9, / 30

12 Theorem (H. 2018) The generating function of I R n (e i e j < e k ) is 2x x 4x 2. 4x Proof continued. Chetak Hossain (NCSU) Combinatorics of OEIS A July 9, / 30

13 Theorem (H. 2018) The generating function of I R n (e i e j < e k ) is 2x x 4x 2. 4x Proof continued. Conversely, given a weakly increasing sequence of length s and a subset of [0, s] sorted decreasing, we can glue them together to form an inversion sequence. Chetak Hossain (NCSU) Combinatorics of OEIS A July 9, / 30

14 Theorem (H. 2018) The generating function of I R n (e i e j < e k ) is 2x x 4x 2. 4x Proof continued. Conversely, given a weakly increasing sequence of length s and a subset of [0, s] sorted decreasing, we can glue them together to form an inversion sequence. I s,k = {((e 1,..., e s ), (e s+1,..., e s+k )) (e 1,..., e s ) C s, s e s+1 > } Chetak Hossain (NCSU) Combinatorics of OEIS A July 9, / 30

15 Theorem (H. 2018) The generating function of I R n (e i e j < e k ) is 2x x 4x 2. 4x Proof continued. Conversely, given a weakly increasing sequence of length s and a subset of [0, s] sorted decreasing, we can glue them together to form an inversion sequence. I s,k = {((e 1,..., e s ), (e s+1,..., e s+k )) (e 1,..., e s ) C s, s e s+1 > } We can count the size of I s,k, by noting that the left piece is counted by a Catalan number, and the right piece is counted by a binomial coefficient. Chetak Hossain (NCSU) Combinatorics of OEIS A July 9, / 30

16 Theorem (H. 2018) The generating function of I R n (e i e j < e k ) is 2x x 4x 2. 4x Proof continued. Conversely, given a weakly increasing sequence of length s and a subset of [0, s] sorted decreasing, we can glue them together to form an inversion sequence. I s,k = {((e 1,..., e s ), (e s+1,..., e s+k )) (e 1,..., e s ) C s, s e s+1 > } We can count the size of I s,k, by noting that the left piece is counted by a Catalan number, and the right piece is counted by a binomial coefficient. ( ) s + 1 I s,k = c s k Chetak Hossain (NCSU) Combinatorics of OEIS A July 9, / 30

17 Theorem (H. 2018) The generating function of I R n (e i e j < e k ) is 2x x 4x 2. 4x Proof continued. Chetak Hossain (NCSU) Combinatorics of OEIS A July 9, / 30

18 Theorem (H. 2018) The generating function of I R n (e i e j < e k ) is 2x x 4x 2. 4x Proof continued. The generating function for twice the number of sequences is essentially the generating function for I s,n s. Chetak Hossain (NCSU) Combinatorics of OEIS A July 9, / 30

19 Theorem (H. 2018) The generating function of I R n (e i e j < e k ) is 2x x 4x 2. 4x Proof continued. The generating function for twice the number of sequences is essentially the generating function for I s,n s. 2 a n x n = n=2 s=2 n=s I s,n s x n + 2x 2 + x 3 Chetak Hossain (NCSU) Combinatorics of OEIS A July 9, / 30

20 Theorem (H. 2018) The generating function of I R n (e i e j < e k ) is 2x x 4x 2. 4x Proof continued. Chetak Hossain (NCSU) Combinatorics of OEIS A July 9, / 30

21 Theorem (H. 2018) The generating function of I R n (e i e j < e k ) is 2x x 4x 2. 4x Proof continued. 2 a n x n = n=2 s=2 n=s k = n s 2 a n x n = c s x s n=2 s=2 I s,n s x n + 2x 2 + x 3 k=0 ( s + 1 k ) x k + 2x 2 + x 3 Chetak Hossain (NCSU) Combinatorics of OEIS A July 9, / 30

22 Theorem (H. 2018) The generating function of I R n (e i e j < e k ) is 2x x 4x 2. 4x Proof continued. By the binomial theorem: 2 a n x n = c s x s (x + 1) s+1 + 2x 2 + x 3 n=2 s=2 We note that by convention, a 0 = a 1 = 1. 2(A(x) 1 x) = c s x s (x + 1) s+1 + 2x 2 + x 3 s=2 Chetak Hossain (NCSU) Combinatorics of OEIS A July 9, / 30

23 Theorem (H. 2018) The generating function of I R n (e i e j < e k ) is 2x x 4x 2. 4x Proof continued. We recall that the Catalan numbers have generating function c s y s = 1 1 4y. Using the Catalan generating function for 2y s=0 y = x(1 + x), and after a routine computation, we find the desired generating function: n=0 a n x n = 2x x(x + 1) 4x Chetak Hossain (NCSU) Combinatorics of OEIS A July 9, / 30

24 Lattice Paths Definition Let D n be the set of Dyck paths, that is, underdiagonal paths in a n n box that use north and east steps of length 1. Chetak Hossain (NCSU) Combinatorics of OEIS A July 9, / 30

25 Lattice Paths Definition Let D n be the set of Dyck paths, that is, underdiagonal paths in a n n box that use north and east steps of length 1. Definition Let SP n be the set of Schröder paths, that is underdiagonal paths in a n n box consisting of north (length 1), east (length 1), and diagonal northeast (length 2) steps. Chetak Hossain (NCSU) Combinatorics of OEIS A July 9, / 30

26 Lattice Paths Definition Let D n be the set of Dyck paths, that is, underdiagonal paths in a n n box that use north and east steps of length 1. Definition Let SP n be the set of Schröder paths, that is underdiagonal paths in a n n box consisting of north (length 1), east (length 1), and diagonal northeast (length 2) steps. Definition Let RSP n SP n be the set of restricted Schröder paths, where there are no diagonal steps on the main diagonal and every diagonal step is immediately followed by an east step. Chetak Hossain (NCSU) Combinatorics of OEIS A July 9, / 30

27 Lattice Paths Definition Let D n be the set of Dyck paths, that is, underdiagonal paths in a n n box that use north and east steps of length 1. Definition Let SP n be the set of Schröder paths, that is underdiagonal paths in a n n box consisting of north (length 1), east (length 1), and diagonal northeast (length 2) steps. Definition Let RSP n SP n be the set of restricted Schröder paths, where there are no diagonal steps on the main diagonal and every diagonal step is immediately followed by an east step. D n RSP n SP n Chetak Hossain (NCSU) Combinatorics of OEIS A July 9, / 30

28 Building Schröder paths from Dyck paths Chetak Hossain (NCSU) Combinatorics of OEIS A July 9, / 30

29 Lattice Path Examples Chetak Hossain (NCSU) Combinatorics of OEIS A July 9, / 30

30 Counting Restricted Schröder Paths Theorem RSP n is counted by OEIS A Chetak Hossain (NCSU) Combinatorics of OEIS A July 9, / 30

31 Counting Restricted Schröder Paths Theorem RSP n is counted by OEIS A [Aguiar-Moreira 2006] showed that a certain family of trees is counted by OEIS A The paths are in bijection with the trees. Chetak Hossain (NCSU) Combinatorics of OEIS A July 9, / 30

32 Pattern Avoiding Permutations Definition S n (4123, 4132, 2413, 3412) is the set of permutations such that for any descent π i π i+1 where π i+1 π i 3, all the values between π i+1 and π i occur to the left of π i. Theorem (H. 2015) S n (4123, 4132, 2413, 3412) is counted by OEIS A Chetak Hossain (NCSU) Combinatorics of OEIS A July 9, / 30

33 Dyck Inversions Chetak Hossain (NCSU) Combinatorics of OEIS A July 9, / 30

34 Dyck Inversions Definition We call a pair (σ i, σ j ) an inversion of σ if i < j and σ i > σ j. Chetak Hossain (NCSU) Combinatorics of OEIS A July 9, / 30

35 Dyck Inversions Definition We call a pair (σ i, σ j ) an inversion of σ if i < j and σ i > σ j. Definition A non-dyck inversion of a permutation w S n is an inversion (σ i, σ j ) such that there exists some σ k where i < j < k and σ j < σ k < σ i. A Dyck inversion of a permutation σ S n is an inversion (σ i, σ j ) that is not a non-dyck inversion. Chetak Hossain (NCSU) Combinatorics of OEIS A July 9, / 30

36 τ example τ(641532) = Chetak Hossain (NCSU) Combinatorics of OEIS A July 9, / 30

37 Using Dyck Inversions to define a map τ : S n D n Definition Let τ : S n D n be the following map. Let d i = {j (σ i, σ j ) is a Dyck inversion} τ(σ) is the unique Dyck path where the east step in the ith column occurs at height i d i + 1. Chetak Hossain (NCSU) Combinatorics of OEIS A July 9, / 30

38 Using Dyck Inversions to define a map τ : S n D n Definition Let τ : S n D n be the following map. Let d i = {j (σ i, σ j ) is a Dyck inversion} τ(σ) is the unique Dyck path where the east step in the ith column occurs at height i d i + 1. τ when restricted to S n (312) recovers τ Av. Chetak Hossain (NCSU) Combinatorics of OEIS A July 9, / 30

39 Using Dyck Inversions to define a map τ : S n D n Definition Let τ : S n D n be the following map. Let d i = {j (σ i, σ j ) is a Dyck inversion} τ(σ) is the unique Dyck path where the east step in the ith column occurs at height i d i + 1. τ when restricted to S n (312) recovers τ Av. Theorem (Bandlow-Killpatrick 2001) The map τ Av is a bijection. Moreover, it is statistic preserving sending inversions to area. Chetak Hossain (NCSU) Combinatorics of OEIS A July 9, / 30

40 τ-fibers Chetak Hossain (NCSU) Combinatorics of OEIS A July 9, / 30

41 τ-posets Chetak Hossain (NCSU) Combinatorics of OEIS A July 9, / 30

42 Properties of τ Chetak Hossain (NCSU) Combinatorics of OEIS A July 9, / 30

43 Properties of τ τ is surjective. Chetak Hossain (NCSU) Combinatorics of OEIS A July 9, / 30

44 Properties of τ τ is surjective. The fibers of τ are intervals in the weak order Chetak Hossain (NCSU) Combinatorics of OEIS A July 9, / 30

45 Properties of τ τ is surjective. The fibers of τ are intervals in the weak order The top elements of the intervals have reverse Lehmer codes that are weakly increasing sequences. Chetak Hossain (NCSU) Combinatorics of OEIS A July 9, / 30

46 Properties of τ τ is surjective. The fibers of τ are intervals in the weak order The top elements of the intervals have reverse Lehmer codes that are weakly increasing sequences. τ restricted to the top elements gives a bijection between the weakly increasing sequences and D n. Chetak Hossain (NCSU) Combinatorics of OEIS A July 9, / 30

47 A map ω : S n RSP n Chetak Hossain (NCSU) Combinatorics of OEIS A July 9, / 30

48 A map ω : S n RSP n Given a permutation σ, build the Dyck path τ(σ). Chetak Hossain (NCSU) Combinatorics of OEIS A July 9, / 30

49 A map ω : S n RSP n Given a permutation σ, build the Dyck path τ(σ). Use the relative order of the atoms of the binary tree in the permutation to decide where the triangles appear. Chetak Hossain (NCSU) Combinatorics of OEIS A July 9, / 30

50 ω examples ω(146532)= ω(164532)= ω(465132)= ω(641532)= Chetak Hossain (NCSU) Combinatorics of OEIS A July 9, / 30

51 ω-fibers Chetak Hossain (NCSU) Combinatorics of OEIS A July 9, / 30

52 ω-fiber codes Chetak Hossain (NCSU) Combinatorics of OEIS A July 9, / 30

53 ω-fiber codes Chetak Hossain (NCSU) Combinatorics of OEIS A July 9, / 30

54 Bijection between Inversion Sequences and RSP n We defined a surjective map ω : S n RSP n Chetak Hossain (NCSU) Combinatorics of OEIS A July 9, / 30

55 Bijection between Inversion Sequences and RSP n We defined a surjective map ω : S n RSP n The fibers of ω are intervals. Chetak Hossain (NCSU) Combinatorics of OEIS A July 9, / 30

56 Bijection between Inversion Sequences and RSP n We defined a surjective map ω : S n RSP n The fibers of ω are intervals. The Lehmer codes of the top elements of the intervals are precisely the pattern avoiding inversion sequences. Chetak Hossain (NCSU) Combinatorics of OEIS A July 9, / 30

57 Bijection between Inversion Sequences and RSP n We defined a surjective map ω : S n RSP n The fibers of ω are intervals. The Lehmer codes of the top elements of the intervals are precisely the pattern avoiding inversion sequences. The number of inversion sequences is the same as the number of paths. Chetak Hossain (NCSU) Combinatorics of OEIS A July 9, / 30

58 Bijection between Inversion Sequences and RSP n We defined a surjective map ω : S n RSP n The fibers of ω are intervals. The Lehmer codes of the top elements of the intervals are precisely the pattern avoiding inversion sequences. The number of inversion sequences is the same as the number of paths. Therefore, ω restricted to the top elements gives the desired bijection. Chetak Hossain (NCSU) Combinatorics of OEIS A July 9, / 30

59 Thank you for listening! Chetak Hossain (NCSU) Combinatorics of OEIS A July 9, / 30