A combinatorial bijection on k-noncrossing partitions

Transcription

1 A combinatorial bijection on k-noncrossing partitions KAIST Korea Advanced Institute of Science and Technology 2018 JMM Special Session in honor of Dennis Stanton January 10, 09:30 09:50

2 I would like to thank Professor Dennis Stanton for teaching me the Pleasure of Doing Combinatorics

3 Abstract For any integer k 2, we prove combinatorially the following Euler (binomial) transformation identity NC (k) n+1 (t) = t n i=0 ( ) n i NW (k) i (t), where NC (k) m (t) (resp. NW (k) m (t)) is the enumerative polynomial on partitions of {1,..., m} avoiding k-crossings (resp. enhanced k-crossings) by number of blocks. The special k = 2 and t = 1 case, asserting the Euler transformation of Motzkin numbers are Catalan numbers, was discovered by Donaghey The result for k = 3 and t = 1, arising naturally in a recent study of pattern avoidance in ascent sequences and inversion sequences, was proved only analytically.

4 Set partitions For a positive integer n, let [n] denote the set {1, 2,..., n}. P = {B 1, B 2,..., B k } is a set partition of [n] with k blocks, if B i s are nonempty subsets of [n], B i s are mutually disjoint, and i B i = [n] Π n : the set of all set partions of [n]. S(n, k), the Stirling number of the second kind, is the number of set partitions of [n] with k blocks.

5 Arc diagram of partitions Nodes are 1, 2,..., n from left to right. There is an arc from i to j, i < j, whenever both i and j belong to a same block, say B P, and B contains no l with i < l < j. There is a loop from i to itself if {i} is a block in P. Example The arc diagram of {{1, 3, 7}, {2, 5, 6}, {4}} Π

6 Crossing A partition has a crossing if there exists two arcs (i 1, j 1 ) and (i 2, j 2 ) in its arc diagram such that i 1 < i 2 < j 1 < j 2. It is well known that the number of partitions in Π n with no crossings is given by the n-th Catalan number C n = 1 ( ) 2n. n + 1 n The crossings of partitions have a natural generalization called k-crossings for any fixed integer k 2.

7 Crossing The arc diagram of {{1, 3, 7}, {2, 5, 6}, {4}} Π

8 Crossing The arc diagram of {{1, 3, 7}, {2, 5, 6}, {4}} Π

9 k-crossing and k-noncrossing A k-crossing of P Π n is a k-subset (i 1, j 1 ), (i 2, j 2 ),..., (i k, j k ) of arcs in the arc diagram of P such that i 1 < i 2 < < i k < j 1 < j 2 < < j k. A partition without any k-crossing is a k-noncrossing partition. A 3-crossing is depicted below:

10 Weak k-crossing and enhanced k-crossing A weak k-crossing of P Π n is a k-subset (i 1, j 1 ), (i 2, j 2 ),..., (i k, j k ) of arcs in the arc diagram of P such that i 1 < i 2 < < i k = j 1 < j 2 < < j k. The k-crossings and weak k-crossings of P are collectively called the enhanced k-crossings of P. A partition without any enhanced k-crossing is an enhanced k-noncrossing partition.

11 Example: Weak 3-crossing and weak 3-nesting A 3-crossing and a weak 3-crossing are depicted below:

12 Review Recall the arc diagram of {{1, 3, 7}, {2, 5, 6}, {4}} Π Two crossings One weak 3-crossing One weak crossing which is not a 3-crossing

13 Review Recall the arc diagram of {{1, 3, 7}, {2, 5, 6}, {4}} Π Two crossings One weak 3-crossing One weak crossing which is not a 3-crossing

14 Review Recall the arc diagram of {{1, 3, 7}, {2, 5, 6}, {4}} Π Two crossings One weak 3-crossing One weak crossing which is not a 3-crossing

15 NC (k) n and NW (k) n Definition Let NC (k) n be the set of all k-noncrossing partitions in Π n. Definition Let NW (k) n in Π n. be the set of all enhanced k-noncrossing partitions If k is sufficently large, i.e. k > n+1 2, then we have NW (k) n = NC (k) n = Π n.

16 NC (k) m (t): Partitions without k-crossings Definition Let NC (k) m (t) be the enumerative polynomial on partitions of [m] avoiding k-crossings by number of blocks. The following contributes t 3 to NC (3) 7 (t)

17 NW (k) m (t): Partitions without enhanced k-crossings Definition Let NW (k) m (t) be the enumerative polynomial on partitions of [m] avoiding enhanced k-crossings by number of blocks. The following contributes t 3 to NW (4) 7 (t)

18 Main result Theorem For n 1 and k 2, n ( ) NC (k) n n+1 (t) = t NW (k) i (t), (1) i i=0 where NW (k) 0 (t) = 1 by convention. The t = 1 case of (1) implies that the D-finiteness (differentiably finite) of the generating function of k-noncrossing partitions is the same as that of the generating function of enhanced k-noncrossing partitions. There are several partial results that lead to the discovery of (1).

19 Partial matchings The k = 2 and t = 1 case of NC (k) n+1 (t) = t n i=0 ( ) n i NW (k) i (t). Enhanced 2-noncrossing partitions in Π n are noncrossing partial matchings of [n], i.e. noncrossing partitions for which the blocks have size one or two. Noncrossing partial matchings of [n] are counted by the n-th Motzkin number M n = n/2 ) i=0 Ci, identity (1) reduces to C n+1 = ( n 2i n i=0 ( ) n M i. (2) i

20 The case k > n+1 2 If k is sufficently large, i.e. k > n+1 2, then we have NW (k) n = NC (k) n = Π n, and NC (k) n+1 (t) = t n i=0 ( ) n i NW (k) i (t) is equivalent to for all m 0, S(n + 1, m + 1) = n i=0 ( ) n S(i, m), i where S(a, b) denotes the Stirling number of the second kind.

21 Case: k = 3 and t = 1 Recall (1): NC (k) n+1 (t) = t n i=0 ( ) n i NW (k) i (t), The first nontrivial case of (1) is when k = 3 and t = 1: Known. The t = 1 case of (1) for general k: Conjectured by Lin.

22 Case: k = 2 and general t Recall (1): NC (k) n+1 (t) = t n i=0 ( ) n i Recall (2): k = 2 and t = 1 in the above. C n+1 = n i=0 ( ) n M i i NW (k) i (t), The k = 2 case of (1), which is a t-extension of (2), seems new: C n+1 (t) = t n i=0 ( ) n M i (t). (3) i C n (t) and M n (t) denote the generating functions of noncrossing partitions of [n] and noncrossing partial matchings of [n].

23 Case: General k and t = 1 Recall our objective: For all k, NC (k) n+1 (t) = t n i=0 ( ) n i NW (k) i (t) For t = 1, the above identity has multiple proofs. But they, except what comes next, do not prove the above as a polynomial in t.

24 Bijective proof for k = 2 and general t We will first illustrate our bijective proof of (1), for k = 2, NC (2) n+1 (t) = t n i=0 ( ) n i NW (2) i (t), for noncrossing partitions, and then extend it to all k-noncrossing partitions. The extension of our construction from k = 2 to general k is highly nontrivial. So we show our framework for the noncrossing partition case first. From now on, we let Π n denote the set of partitions of {0, 1,..., n 1} rather than partitions of [n], for convenience s sake.

25 Bijective proof for k = 2 and general t We give a combinatorial interpretation of identity (3), C n+1 (t) = t n i=0 First we interpret the right hand side t n i=0 ( ) n M i (t). i ( ) n M i (t) i as the generating function of all pairs (A, µ) such that A is a subset of {1, 2,..., n} and µ is a noncrossing matching whose nodes are elements of A placed on the line in the natural order. A pair (A, µ) is weighted by t µ +1, where µ is the number of blocks of µ. If A is the empty set, then µ is the empty matching with weight t.

26 Bijection Ψ for k = 2 and general t We now define a combinatorial bijection Ψ from noncrossing partitions in Π n+1 to the set of all pairs (A, µ) in the above. Let n = 10 and π = {{0, 8, 10}, {1, 2, 7}, {3, 5, 6}, {4}, {9}}. This π is a noncrossing partition as can be seen below:

27 Bijection Ψ for k = 2 and general t Consider all blocks in π which do not contain 0: {{1, 2, 7}, {3, 5, 6}, {4}, {9}}. From each block, delete all integers which are neither the smallest nor the largest in the block. Let the resulting set be µ, and let A be the union of all blocks in µ: (A, µ) = ({1, 3, 4, 6, 7, 9}, {{1, 7}, {3, 6}, {4}, {9}}) The next figure shows the elements of A, in blue, and the matching µ Let Ψ(π) = (A, µ). Clearly, this is weight-preserving.

28 Example: Ψ Delete arcs connected to 0.

29 Example: Ψ Delete arcs connected to 0.

30 Example: Ψ Delete arcs connected to 0. Let go of 2 and 5.

31 Example: Ψ Delete arcs connected to 0. Let go of 2 and 5.

32 Example: Ψ Let Ψ(π) = (A, µ). Clearly, this is weight-preserving.

33 Bijection Ψ 1 for k = 2 and general t The above procedure is reversible. Let (A, µ) be a pair such that A is a subset of {1, 2,..., n} and µ is a noncrossing matching whose nodes are elements of A placed on the line in the natural order. We will construct the corresponding partition π of {0, 1, 2,..., n} as follows. Interpret each block β in µ as an interval I (β) = {i : min{β} i max{β}}. Let the block of π containing 0 be {0, 1, 2,..., n} \ β µ I (β).

34 Example: Ψ 1 As an example, let n = 10 and (A, µ) = ({1, 3, 4, 6, 7, 9}, {{1, 7}, {3, 6}, {4}, {9}}). The block containing 0 is {0, 8, 10}, shown in red below

35 Example: Ψ 1 Other blocks of π are obtained by extending blocks in µ by the rule: i {1, 2,..., n} \ A belongs to the block originating from a block β µ if I (β) is the smallest interval containing i. In our example, two blocks {1, 7} and {3, 6} are enlarged, shown in blue below So Ψ 1 (π) = {{0, 8, 10}, {1, 2, 7}, {3, 5, 6}, {4}, {9}}.

36 Bijection Ψ Determine the block containing 0. Pick up 2 and 5. So Ψ 1 (π) = {{0, 8, 10}, {1, 2, 7}, {3, 5, 6}, {4}, {9}}.

37 Bijection Ψ Determine the block containing 0. Pick up 2 and 5. So Ψ 1 (π) = {{0, 8, 10}, {1, 2, 7}, {3, 5, 6}, {4}, {9}}.

38 Bijection Ψ Determine the block containing 0. Pick up 2 and 5. So Ψ 1 (π) = {{0, 8, 10}, {1, 2, 7}, {3, 5, 6}, {4}, {9}}.

39 Red block, colored arc diagram Definition (Red block, colored arc diagram) In a partition P, the block containing 0 is called a red block, denoted by red(p), and other blocks are called black blocks. The elements in red(p) are colored red, and other elements are colored black. Arcs in arc diagram of P between red elements are colored red and other arcs are colored black. Such a colored version of arc diagram of P is called the colored arc diagram, denoted by D(P). A partition is called k-crossing if it has at least one k-crossing. A (weak) k-crossing is called a black (weak) k-crossing, if all its arcs are black; a red (weak) k-crossing, otherwise.

40 NC (k) n, NW (k), NBW (k) n n Recall: NC (k) n is the set of all k-noncrossing partitions in Π n. NW (k) n is the set of all partitions in Π n which avoid enhanced k-crossings, i.e., have neither k-crossings nor weak k-crossings. (Enhanced k-noncrossing) Definition Let NBW (k) n be the set of all partitions P in Π n whose colored arc diagram, D(P), has neither black k-crossings nor black weak k-crossings.

41 An example in NBW (3) 17 P NBW (3) 17 and its colored diagram D(P): P = {{0, 4, 8, 15}, {1, 3, 10}, {2, 11}, {5, 16}, {6, 13}, {7, 9, 12, 14}} There are no black 3-crossings and no black weak 3-crossings. There are red 3-crossings and red weak 3-crossings.

42 Decomposition of NBW (k) n For any subset A of {1, 2,..., n 1}, define a subset Π A of Π n = Π {0,1,...,n 1} by Π A = {P Π n : red(p) = {0, 1,..., n 1} \ A}. Π n is partitioned into {Π A } A {1,2,...,n 1}, and there is a natural correspondence between Π A and Π A. If A = {a 1, a 2,, a l } with a 1 < a 2 < < a l then the correspondence is obtained by mapping a i to i 1 for each i. This correspondence reduces the number of blocks by 1, since the red block is ignored.

43 Decomposition of NBW (k) n Let us define a subset NBW (k) A NBW (k) A of NBW(k) n by = Π A NBW (k) n. We can see that NBW (k) n is partitioned into {NBW (k) A } A {1,2,...,n 1}, and there is a natural correspondence between NBW (k) A and, i.e., the restriction of the natural correspondence between NW (k) A Π A and Π A.

44 NC (k) n+1 and NBW(k) n+1 Define a weight function w on Π n by w(p) = t P for each P Π n, where P denotes the number of blocks in P. Since we have w(p) = NC (k) n+1 (t) and P NBW (k) n+1 P NC (k) n+1 w(p) = = = t A {1,...,n} P NBW (k) A A {1,...,n} n i=0 t ( ) n i P NW (k) A NW (k) i (t), identity (1) is equivalent to the following theorem. w(p) w(p)

45 Combinatorial bijection Φ Theorem For all n and k, there exists a weight-preserving combinatorial bijection Φ : NBW (k) n+1 NC(k) n+1 proving w(p) = w(p). P NBW (k) n+1 P NC (k) n+1

46 What are we looking for? Recall our earlier example, P NBW (3) 17, P = {{0, 4, 8, 15}, {1, 3, 10}, {2, 11}, {5, 16}, {6, 13}, {7, 9, 12, 14}} There are no black 3-crossings and no black weak 3-crossings. There are red 3-crossings and red weak 3-crossings We need to find Φ(P) NC (3) 17

47 The center of a k-crossing Recall that a weak k-crossing of P Π n is a k-subset (i 1, j 1 ), (i 2, j 2 ),..., (i k, j k ) of arcs such that i 1 < i 2 < < i k = j 1 < j 2 < < j k. The position c, c = i k = j 1, is called the center of the weak k-crossing. We will say that a node α is under a k-crossing (i 1, j 1 ), (i 2, j 2 ),..., (i k, j k ), if i k < α < j 1, i.e. i 1 < i 2 < < i k < α < j 1 < j 2 < < j k.

48 Combinatorial bijection Φ Theorem For all n and k, there exists a weight-preserving combinatorial bijection Φ : NBW (k) n+1 NC(k) n+1 proving w(p) = w(p). Outline: P NBW (k) n+1 P NC (k) n+1 1 Change red nodes under black (k 1)-crossing into centers of black weak k-crossings. 2 Change red k-crossings into red nodes under black (k 1)-crossings.

49 Combinatorial bijection Φ An example of Φ : NBW (3) 17 NC(3) 17 with (n, k) = (16, 3): P = {{0, 4, 8, 15}, {1, 3, 10}, {2, 11}, {5, 16}, {6, 13}, {7, 9, 12, 14}} Red 8 is under four black 2-crossings of which the innermost is (3, 10), (6, 13). Change 8 to the center of a black weak 3-crossing, (3, 8), (6, 10), (8, 13). Uncolor 8.

50 Combinatorial bijection Φ An example of Φ : NBW (3) 17 NC(3) 17 with (n, k) = (16, 3): P = {{0, 4, 8, 15}, {1, 3, 10}, {2, 11}, {5, 16}, {6, 13}, {7, 9, 12, 14}} Red 8 is under four black 2-crossings of which the innermost is (3, 10), (6, 13).

51 Combinatorial bijection Φ An example of Φ : NBW (3) 17 NC(3) 17 with (n, k) = (16, 3): P = {{0, 4, 8, 15}, {1, 3, 10}, {2, 11}, {5, 16}, {6, 13}, {7, 9, 12, 14}} Red 8 is under four black 2-crossings of which the innermost is (3, 10), (6, 13). Change 8 to the center of a black weak 3-crossing, (3, 8), (6, 10), (8, 13). Uncolor 8, and fix colors.

52 Combinatorial bijection Φ Blue indicates the weak 3-crossing.

53 Combinatorial bijection Φ Blue indicates the weak 3-crossing.

54 Combinatorial bijection Φ Arcs (2, 11), (4, 15), (5, 16) form a red 3-crossing. Do cyclic rotation : (2, 11), (4, 15), (5, 16) (2, 15), (4, 11), (5, 16) (i 1, j 1 ), (i 2, j 2 ),..., (i p 1, j p ), (i p, j p ), (i p+1, j p+1 ),..., (i k, j k ) (i 1, j 2 ), (i 2, j 3 ),..., (i p 1, j p ), (i p, j 1 ), (i p+1, j p+1 ),..., (i k, j k )

55 Combinatorial bijection Φ Arcs (2, 11), (4, 15), (5, 16) form a red 3-crossing. Do cyclic rotation : (2, 11), (4, 15), (5, 16) (2, 15), (4, 11), (5, 16)

56 Combinatorial bijection Φ Arcs (3, 8), (4, 11), (5, 16) form a red 3-crossing. Do cyclic rotation : (3, 8), (4, 11), (5, 16) (3, 11), (4, 8), (5, 16) (i 1, j 1 ), (i 2, j 2 ),..., (i p 1, j p ), (i p, j p ), (i p+1, j p+1 ),..., (i k, j k ) (i 1, j 2 ), (i 2, j 3 ),..., (i p 1, j p ), (i p, j 1 ), (i p+1, j p+1 ),..., (i k, j k )

57 Combinatorial bijection Φ Arcs (3, 8), (4, 11), (5, 16) form a red 3-crossing. Do cyclic rotation : (3, 8), (4, 11), (5, 16) (3, 11), (4, 8), (5, 16)

58 Combinatorial bijection Φ The last colored arc diagram corresponds to Φ(P) NC (3) 17 : P = {{0, 4, 8, 15}, {1, 3, 10}, {2, 11}, {5, 16}, {6, 13}, {7, 9, 12, 14}}. Φ(P) = ({0, 4, 8, 13}, {1, 3, 11}, {2, 15}, {5, 16}, {6, 10}, {7, 9, 12, 14}).

59 Combinatorial bijection Φ 1 The crucial reason why Φ is reversible is that whenever we do the cyclic rotation to a red k-crossing, we create a red node under a black (k 1)-crossing. In the following, we show Φ is a bijection by defining its inverse explicitly: Φ 1 : NC (k) n+1 NBW(k) n+1.

60 Combinatorial bijection Φ 1 An example of Φ 1 with (n, k) = (16, 4) and P NC (4) 17 : P = {{0, 3, 5, 14}, {1, 6, 8, 11, 12}, {2, 9, 15}, {4, 10, 16}, {7, 13}} Red 5 is under a black 3-crossing: (1, 6), (2, 9), (4, 10). (1, 6), (2, 9), (3, 5), (4, 10) (1, 5), (2, 6), (3, 9), (4, 10) (i 1, j 1 ), (i 2, j 2 ),..., (i t, j t ), (a, a), (i t+1, j t+1 ),..., (i k 1, j k 1 ) (i 1, a), (i 2, j 1 ),..., (i t, j t 1 ), (a, j t ), (i t+1, j t+1 ),..., (i k 1, j k 1 )

61 Combinatorial bijection Φ 1 Red 5 is under a black 3-crossing: (1, 6), (2, 9), (4, 10). (1, 6), (2, 9), (3, 5), (4, 10) (1, 5), (2, 6), (3, 9), (4, 10)

62 Combinatorial bijection Φ Arcs (1, 5), (2, 6), (4, 10), (5, 14) form a black weak 4-crossing. Change them into a black 3-crossing (1, 6), (2, 10), (4, 14).

63 Combinatorial bijection Φ Change black weak 4-crossing (1, 5), (2, 6), (4, 10), (5, 14) into a black 3-crossing (1, 6), (2, 10), (4, 14) The last colored arc diagram corresponds to Φ 1 (P) NBW (4) 17 : Φ 1 (P) = {{0, 3, 5, 9, 15}, {1, 6, 8, 11, 12}, {2, 10, 16}, {4, 14}, {7, 13}}.

64 Problem Is there any generating function approach to (1)? NC (k) n+1 (t) = t n i=0 ( ) n i NW (k) i (t)

65 Reference This presentation is based on a preprint,, by Zhicong Lin and, which will be posted soon.

66 Thank you very much Department of Mathematical Sciences KAIST

67 I would like to thank Professor Dennis Stanton for teaching me the Pleasure of Doing Combinatorics