A cyclic sieving phenomenon in Catalan Combinatorics

Transcription

1 A cyclic sieving phenomenon in Catalan Combinatorics Christian Stump Université du Québec à Montréal SLC 64, Lyon March 30, 2010 Christian Stump A cyclic sieving phenomenon in Catalan Combinatorics LaCIM, UQAM, Montréal 1 of 15

2 Non-crossing partitions Non-crossing partitions can be defined for any Coxeter group W as NC(W,c) := { ω W : l T (ω) + l T (cω 1 ) = l T (c) }, where c is a Coxeter element and where l T (ω) denotes the absolute length on W, for W = S n being the symmetric group, NC(S n ) = {ω = c 1...c k S n : c i increasing; c i,c j non-crossing}, where the Coxeter element c = (1,..., n) and where ω = c 1...c k is the cycle notation of ω. We focus on S n (and keep the general case in mind). Christian Stump A cyclic sieving phenomenon in Catalan Combinatorics LaCIM, UQAM, Montréal 2 of 15

3 Example: NC(S 4 ) (1342) (1243) (1324) (1234) (1432) (1423) (132) (243) (13)(24) (123) (234) (134) (124) (12)(34) (14)(23) (143) (142) 3214 (13) 1432 (24) 2134 (12) 1324 (23) 1243 (34) 4231 (14) 1234 Christian Stump A cyclic sieving phenomenon in Catalan Combinatorics 1 LaCIM, UQAM, Montréal 3 of 15

4 Non-nesting partitions Non-nesting partitions can be defined for any crystallographic Coxeter group W as NC(W ) := {A Φ + : A antichain}, where Φ + denotes the root poset of W, for W = S n being the symmetric group, NN(S n ) is the set of all antichains in Christian Stump A cyclic sieving phenomenon in Catalan Combinatorics LaCIM, UQAM, Montréal 4 of 15

5 Example: NN(S 4 ),[12],[23],[34],[13],[24],[14], [12] [23],[12] [34],[23] [34],[12] [24], [13] [34],[13] [24],[12] [23] [34] Christian Stump A cyclic sieving phenomenon in Catalan Combinatorics LaCIM, UQAM, Montréal 5 of 15

6 Non-crossing and non-nesting partitions Theorem (Athanasiadis 2004) For any crystallographic Coxeter group W of rank l, the number of non-crossing and of non-nesting partitions coincide with the W -Catalan number, l NC(W ) = NN(W ) = Cat(W ) := i=1 d i + h d i, where d 1... d l denote the degrees of the fundamental invariants of W. Christian Stump A cyclic sieving phenomenon in Catalan Combinatorics LaCIM, UQAM, Montréal 6 of 15

7 An important open problem Different groups found explicit bijections between non-crossing and non-nesting partitions for various types, but the general connection is still open: Open Problem Find a type-independent bijection between NC(W ) and NN(W ). Christian Stump A cyclic sieving phenomenon in Catalan Combinatorics LaCIM, UQAM, Montréal 7 of 15

8 Further refinements Define a cyclic group action on NC(W ) by Kreweras complementation, K(ω) := c ω 1. Observe that K 2 (ω) = c ω c 1 is conjugation by c. Theorem (CSP on non-crossing partitions) { ω NC(W ) : K k (ω) = ω } = Cat(W ;ζ k d ) := l i=1 [d i + h] q [d i ] q, q=ζ k d where the product Cat(W ;q) is a q-analogue of the W -Catalan number Cat(W ) and where d = 2h is the order of the cyclic group. Christian Stump A cyclic sieving phenomenon in Catalan Combinatorics LaCIM, UQAM, Montréal 8 of 15

9 Further refinements The theorem is an instance of the cyclic sieving phenomenon (which you all know from the SLC 62 one year ago in Heilsbronn, Germany). It was proved accidentally for the symmetric group by C. Heitsch, and it was recently proved in much more generality by C. Krattenthaler in an unpublished manuscript and by him together with T. Müller in two articles on 134 pages, it generalizes a theorem by D. Bessis and V. Reiner on the CSP by conjugation. Christian Stump A cyclic sieving phenomenon in Catalan Combinatorics LaCIM, UQAM, Montréal 9 of 15

10 Example (continued): NC(S 4 ) (12) (1234)(12) = (134) (1234)(143) = (23) (1234)(23) = (124) (1234)(142) = (34) (1234)(34) = (123) (1234)(132) = (14) (1234)(14) = (234) (1234)(243) = (12) (13) (1234)(13) = (14)(23) (1234)(14)(23) = (24) (1234)(24) = (12)(34) (1234)(12)(34) = (13) () (1234)() = (1234) (1234)(4321) = () Cat(S n ;q) = 1+q 2 +q 3 +2q 4 +q 5 +2q 6 +q 7 +2q 8 +q 9 +q 10 +q 12 evaluated at 8-th roots of unity gives 14 = NC if ζ = 1 6 = NC K4 if ζ = 1 2 = NC K2 = NC K6 if ζ = ±i 0 = NC K = NC K3 = NC K5 = NC K7 otherwise Christian Stump A cyclic sieving phenomenon in Catalan Combinatorics LaCIM, UQAM, Montréal 10 of 15

11 Further refinements Define a cyclic group action on NC(W ) by Panyushev complementation P(A) := min{t Φ + : t a for some a A} NN(W ). Conjecture (V. Reiner, SLC 62) { ω NN(W ) : P k (ω) = ω } = Cat(W ;ζ k d ). Or equivalently, there exists a bijection ψ : NN(W ) NC(W ) such that ψ P = K ψ. Christian Stump A cyclic sieving phenomenon in Catalan Combinatorics LaCIM, UQAM, Montréal 11 of 15

12

13 12

14 12 23,34

15 12 23,34

16 12 23,34 12,24

17 12 23,34 12,24

18 12 23,34 12,24 13

19 12 23,34 12,24 13

20 12 23,34 12,

21 12 23,34 12,

22 12 23,34 12, ,23

23 12 23,34 12, ,23

24 12 23,34 12, ,23 13,34

25 12 23,34 12, ,23 13,34

26 12 23,34 12, ,23 13,34 24

27 12 23,34 12, ,23 13,34 24

28 12 23,34 12, ,23 13,

29 12 23,34 12, ,23 13,

30 12 23,34 12, ,23 13,

31 12 23,34 12, ,23 13, ,23,34

32 12 23,34 12, ,23 13, ,23,34 13,24

33 12 23,34 12, ,23 13, ,23,34 13,24

34 12 23,34 12, ,23 13, ,23,34 13,24 14

35 12 23,34 12, ,23 13, ,23,34 13,24 14

36 12 23,34 12, ,23 13, ,23,34 13,24 14

37 12 23,34 12, ,23 13, ,23,34 13, ,23,34

38 12 23,34 12, ,23 13, ,23,34 13, ,23,34

39 12 23,34 12, ,23 13, ,23,34 13, ,23,34 12,34

40 12 23,34 12, ,23 13, ,23,34 13, ,23,34 12,34 23

41 12 23,34 12, ,23 13, ,23,34 13, ,23,34 12,34 23

42 12 23,34 12, ,23 13, ,23,34 13, ,23,34 12, ,34

43 12 23,34 12, ,23 13, ,23,34 13, ,23,34 12, ,34

44 12 23,34 12, ,23 13, ,23,34 13, ,23,34 12, ,34

45 The type A: Non-crossing handshake configurations T n set of non-crossing handshake configurations of 2n, C 2n -action on T n by cyclic permutation of {1,...,2n}. Theorem ( T n,cat(s n ;q), C 2n ) exhibits the CSP. We can construct a bijection ψ 1 : T n NC(S n ), such that ψ 1 c = K ψ 1, and a bijection such that ψ 2 P = c ψ 2. ψ 2 : NN(S n ) T n, The construction can be easily generalized to type B. Christian Stump A cyclic sieving phenomenon in Catalan Combinatorics LaCIM, UQAM, Montréal 13 of 15

46 Towards a uniform bijection The bijection in type A can be generalized to type B, but so far not to type D, the exceptional types can be checked by computer. What does this have to do with a uniform bijection? Christian Stump A cyclic sieving phenomenon in Catalan Combinatorics LaCIM, UQAM, Montréal 14 of 15

47 Towards a uniform bijection Theorem (Conjectured by D. Armstrong in all types) Let L,R be a bipartition of the simple transpositions such that L and R pairwise commute, and let c be the associated bipartite Coxeter element. ψ = ψ 1 ψ 2 : NN(S n ) NC(S n,c = c L c R ) is the unique bijection with the following inductive property: ψ : c L, ψ P = K ψ, ψ(i) = s s L\supp I ψ supp I (I). Remark The theorem gives an inductively defined uniform definition of a bijection between non-nesting and non-crossing partitions in types A and B. Christian Stump A cyclic sieving phenomenon in Catalan Combinatorics LaCIM, UQAM, Montréal 15 of 15