Combinatorics of non-associative binary operations

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1 Combinatorics of non-associative binary operations Jia Huang University of Nebraska at Kearney address: December 26, 2017 This is joint work with Nickolas Hein (Benedictine College), Madison Mickey (UNK) and Jianbai Xu (UNK) Jia Huang (UNK) Combinatorics of non-associative binary operations December 26, / 28

2 Nonassociativity of binary operations Let be a binary operation on a set X. Let x 0, x 1,..., x n be X -valued indeterminates. If is associative then the expression x 0 x 1 x n is unambiguous. Example: x 0 + x x n. If is nonassociative then x 0 x 1 x n depends on parentheses. ((x 0 x 1 ) x 2 ) x 3 = x 0 x 1 x 2 x 3 (x 0 x 1 ) (x 2 x 3 ) = x 0 x 1 x 2 + x 3 (x 0 (x 1 x 2 )) x 3 = x 0 x 1 + x 2 x 3 x 0 ((x 1 x 2 ) x 3 ) = x 0 x 1 + x 2 + x 3 x 0 (x 1 (x 2 x 3 )) = x 0 x 1 + x 2 x 3 The number of ways to parenthesize x 0 x 1 x n is the Catalan number C n := n+1( 1 2n ) n, e.g., (Cn ) 6 n=0 = (1, 1, 2, 5, 14, 42, 132). How many distinct results can be obtained from x 0 x 1 x n? Jia Huang (UNK) Combinatorics of non-associative binary operations December 26, / 28

3 Nonassocitivity measurements Parenthesizations of x 0 x 1 x n are equivalent if they give the same function from X n+1 to X. Call this (, n)-equivalence relation. Define C,n to be the number of (, n)-equivalence classes. Define C,n to be the largest size of (, n)-equivalence classes. ((x 0 x 1 ) x 2 ) x 3 = x 0 x 1 x 2 x 3 C (x 0 x 1 ) (x 2 x 3 )= x 0 x 1 x 2 + x 3 = 5 3 (x 0 (x 1 x 2 )) x 3 = x 0 x 1 + x 2 x 3 C,3 = 4 x 0 ((x 1 x 2 ) x 3 )= x 0 x 1 + x 2 + x 3 C x 0 (x 1 (x 2 x 3 ))= x 0 x 1 + x 2 x,3 = 2 3 Observation In general, 1 C,n C n and 1 C,n C n. C,n = 1, n 0 is associative C,n = C n, n 0. Thus C,n and C,n measure how far is away from being associative. Jia Huang (UNK) Combinatorics of non-associative binary operations December 26, / 28

4 Binary trees Fact Parenthesizations of x 0 x 1 x n (full) binary trees with n + 1 leaves Example ((x 0 x 1 ) x 2 ) x 3 (x 0 (x 1 x 2 )) x 3 (x 0 x 1 ) (x 2 x 3 ) x 0 ((x 1 x 2 ) x 3 ) x 0 (x 1 (x 2 x 3 )) δ = (3, 2, 1, 0) δ = (2, 2, 1, 0) δ = (2, 1, 1, 0) δ = (1, 2, 1, 0) δ = (1, 1, 1, 0) ρ = (0, 1, 1, 1) ρ = (0, 1, 2, 1) ρ = (0, 1, 1, 2) ρ = (0, 1, 2, 2) ρ = (0, 1, 2, 3) Definition The left depth δ i (t) (or right depth ρ i (t)) of leaf i in t T n is the number of edges to the left (right) in the unique path from the root of t down to i. Jia Huang (UNK) Combinatorics of non-associative binary operations December 26, / 28

5 A generalization of associativity Definition A binary operation is k-associative if (x 0 x k ) x k+1 = x 0 (x 1 x k+1 ) where the operations in parentheses are performed left to right. Write C k,n := C,n (k-modular Catalan number) and C k,n := C,n for any operation satisfying precisely the k-associativity. Example (Generalization of + (k = 1) and (k = 2)) Let ω := e 2πi/k be a primitive kth root of unity. Then is k-associative if a b := ωa + b, a, b C. Observation (k = 1: Tamari order) The k-associativity gives the k-associative order on binary trees. Jia Huang (UNK) Combinatorics of non-associative binary operations December 26, / 28

6 Tamari order and 2-associative order on T 4 Jia Huang (UNK) Combinatorics of non-associative binary operations December 26, / 28

7 Components of k-associative order Example (comb 4 and comb 1 4) Theorem (Hein and H. 2017) A binary tree is maximal (or minimal) in the k-associative order if and only if it avoids the binary tree comb k+1 (or comb 1 k ) as a subtree. Each component in k-associative order has a unique minimal tree. Theorem (Hein and H. 2017) Two binary trees t and t correspond to equivalent parenthesizations if and only if δ i (t) δ i (t ) (mod k) for all i. Jia Huang (UNK) Combinatorics of non-associative binary operations December 26, / 28

8 Connections to other objects Fact There are well-known bijections among many families of Catalan objects. Proposition (Hein and H. 2017) For n 0 and k 1, C k,n enumerates the following: 1 the set of binary trees with n + 1 leaves avoiding comb 1 k, 2 plane trees with n non-root nodes, each of degree less than k, 3 Dyck paths of length 2n avoiding DU k (a down-step immediately followed by k up-steps), 4 partitions bounded by (n 1, n 2,..., 1, 0) with each positive part occurring fewer than k times, 5 2 n standard Young tableaux which contain no list of k consecutive numbers in the top row other than 1, 2,..., l for any l [n], 6 permutations of [n] avoiding and 23 (k + 1)1. Jia Huang (UNK) Combinatorics of non-associative binary operations December 26, / 28

9 Examples of Catalan objects ((x 1 x 2 ) x 3 ) x 4 (x 1 x 2 ) (x 3 x 4 ) x 1 ((x 2 x 3 ) x 4 ) x 1 (x 2 (x 3 x 4 )) (x 1 (x 2 x 3 )) x Jia Huang (UNK) Combinatorics of non-associative binary operations December 26, / 28

10 Formulas for C k,n and C k,n Theorem (Hein and H. 2017) For k, n 1, we have C k,n = λ (k 1) n λ <n n λ m λ (1 n ) = n C k,n = 0 j n/k 0 j (n 1)/k n jk n ( 1) j n ( n + j 1 ( n j )( 2n jk n + 1 Moreover, the number of components in k-associative order with size C k,n is C m, where m is the least positive integer congruent to n modulo k. Proof. One proof uses generating functions and Lagrange inversion. The other proof is more direct, using Dyck paths (and sign-reversing involutions). j ). ), Jia Huang (UNK) Combinatorics of non-associative binary operations December 26, / 28

11 Tamari order and 2-associative order on T 4 Jia Huang (UNK) Combinatorics of non-associative binary operations December 26, / 28

12 Modular Catalan numbers Example (C k,n for n 10 and k 8) n C 1,n A C 2,n A C 3,n A C 4,n A C 5,n new C 6,n new C 7,n new C 8,n new C n A Question lim n C n+1 /C n = 4, lim n C k,n+1 /C k,n =? There is a formula C 3,n = ( n 1 )( i 0 i n 1 i i/2 ) obtained by Gouyou-Beauchamps and Viennot in studies of directed animals, and Panyushev using affine Weyl group of the Lie algebra sp 2n or so 2n+1. Is there a generalization of this formula from k = 3 to k 4? Jia Huang (UNK) Combinatorics of non-associative binary operations December 26, / 28

13 Double Minus Definition Define a b := ωa + ηb for a, b C, where ω := e 2πi/k and η := ω 2πi/l. If k = l = 2 we have a b := a b. Let C,n,r be the number of distinct results from x 0 x 1 x n with exactly r plus signs. Let C,n := 0 r n+1 C,n,r. Theorem (H., Mickey, and Xu 2017) If n 1 and 0 r n + 1 then ( n+1 ) r, if n + r 1 (mod 3) and n 2r 2, ( C,n,r = n+1 ) r 1, if n + r 1 (mod 3) and n = 2r 2, 0, if n + r 1 (mod 3). For n 1 we have C,n = { 2 n+1 1 3, if n is odd; 2 n+1 2 3, if n is even. Jia Huang (UNK) Combinatorics of non-associative binary operations December 26, / 28

14 A truncated/modified Pascal Triangle Example (C,n,r for n 10 and 0 r n + 1) r C,0,r 1 C,1,r 1 C,2,r 2 C,3,r 4 1 C,4,r 1 9 C,5,r 15 6 C,6,r C,7,r C,8,r C,9,r C,10,r Jia Huang (UNK) Combinatorics of non-associative binary operations December 26, / 28

15 Double Minus Definition Define a b := ωa + ηb for a, b C, where ω := e 2πi/k and η := ω 2πi/l. If k = l = 2 we have a b := a b. Let C,n,r be the number of distinct results from x 0 x 1 x n with exactly r plus signs. Let C,n := 0 r n+1 C,n,r. Theorem (H., Mickey, and Xu 2017) If n 1 and 0 r n + 1 then ( n+1 ) r, if n + r 1 (mod 3) and n 2r 2, ( C,n,r = n+1 ) r 1, if n + r 1 (mod 3) and n = 2r 2, 0, if n + r 1 (mod 3). For n 1 we have C,n = { 2 n+1 1 3, if n is odd; 2 n+1 2 3, if n is even. Jia Huang (UNK) Combinatorics of non-associative binary operations December 26, / 28

16 OEIS A Definition The sequence A (A n : n 1) = (1, 2, 5, 10, 21, 42, 85,...) has many equivalent characterizations, such as the following. A 1 = 1, A n+1 = 2A n if n is odd, and A n+1 = 2A n + 1 if n is even. A n is the integer with an alternating binary representation of length n. (1 = 1 2, 2 = 10 2, 5 = 101 2, 10 = , 21 = ,...) { 2 n+1 1 A n = 2 n+1 3 = 2n+2 3 ( 1) n 3, if n is odd; 6 = 2 n+1 2 3, if n is even. A n is the number of moves to solve the n-ring Chinese Rings puzzle. n = 4: Question Are there natural bijections between distinct results from parenthesizing x 0 x 1 x n and any other family of objects enumerated by A n? Jia Huang (UNK) Combinatorics of non-associative binary operations December 26, / 28

17 Another Generalization Definition Let C d,e k,l,n := C,n be the number of distinct results obtained from parenthesizing x 0 x 1 x n, where is defined as f g := xf + yg, f, g C[x, y]/(x d+k x d, y e+l y e ) Observation A parenthesization of f 0 f n corresponding to t T n equals x δ0(t) y ρ 0 (t) f x δn(t) y ρ n (t) f n. So one can study C d,e k,l,n using the leaf depths in binary trees. Remark We have results on C d,e k,l,n when two or three of the parameters k, l, d, e are set to be one. In the remainder of this talk, we focus on C d,1 1,1,n and discuss its connections with the algebra of upper triangular matrices. Jia Huang (UNK) Combinatorics of non-associative binary operations December 26, / 28

18 Ideals of upper triangular matrices Definition Let U n be the algebra of all n-by-n upper triangular matrices where a star is an arbitrary entry from a fixed field F (e.g., R). A (two-sided) ideal I of U n is a vector subspace of U n such that XI I and IX I for all X U n. A ideal I is nilpotent if I k = 0 for some k 1. The smallest k such that I k = 0 is the (nilpotent) order of I. A ideal I of U n is commutative if AB = BA for all A, B I. Jia Huang (UNK) Combinatorics of non-associative binary operations December 26, / 28

19 Nilpotent ideals Example (A nilpotent ideal of U 6 and its corresponding Dyck path) I = height = 3 Observation An nilpotent ideal of U n is represented by a matrix of 0 s and s separated by a Dyck path of length 2n. The number of such ideals is the Catalan number C n := 1 n+1( 2n n ). The number of all ideals of U n is the Catalan number C n+1. Jia Huang (UNK) Combinatorics of non-associative binary operations December 26, / 28

20 Commutative ideals Proposition (L. Shapiro, 1975) The number of commutative ideals of U n is 2 n 1. Problem Find a direct proof of the above result. Example (What is a direct proof?) The number of subsets of {1, 2,..., n} is ( n 0) + ( n 1) + ( n 2) + + ( n n) = 2 n. This can be proved directly by considering if a subset contains i for each i. Observation An ideal of U n is commutative if and only if it has nilpotent order 1 or 2. Definition Let C d n be the number of nilpotent ideals of U n with order at most d. Jia Huang (UNK) Combinatorics of non-associative binary operations December 26, / 28

21 Nilpotent order Observation The order of a nilpotent ideal I of U n is the largest possible length d of an admissible sequence, that is, a sequence (i 1, i 2,..., i d ) such that the entry (i j, i j+1 ) is a star in the matrix form of I for all j = 1, 2,..., d 1. Example The following ideal has nilpotent order is 4 since the sequence (1, 3, 5, 6) is admissible and there is no longer admissible sequence I = Jia Huang (UNK) Combinatorics of non-associative binary operations December 26, / 28

22 Bounce Paths Observation Let I be an ideal of U n corresponding to a Dyck path D. Then the nilpotent order of I is the number of times the bounce path of D bounces off the main diagonal. Example (Bounce Path) The bounce path has 4 bounces. The Dyck path D has height 3. Fact (Andrews Krattenthaler Orsina Papi 2002, Haglund 2008) Bijection ζ : Dyck paths with height d Dyck paths with d bounces. Jia Huang (UNK) Combinatorics of non-associative binary operations December 26, / 28

23 Generalization of Commutative Ideals Theorem (H.-Rhoades) Dyck paths of length 2n with height at most d are counted by C d n. Hence C d n is the sequence A in OEIS and interpolates between 1 and C n. Example n n Cn Cn n 1 Cn F 2n 1 Cn (1 + 3n 1 ) 1 C n n ) n+1( n Jia Huang (UNK) Combinatorics of non-associative binary operations December 26, / 28

24 Ideals of Lie Algebras Definition Let sl n (C) be the (type A semisimple) Lie algebra of all n n complex matrices with zero trace under the Lie bracket [X, Y ] := XY YX. Let b be the Borel subalgebra of upper triangular matrices of sl n (C). Theorem (Andrews Krattenthaler Orsina Papi 2002) The number of ad-nilpotent ideals of b with order at most d 1 is C d n. Problem Find a natural order-preserving bijection between nilpotent ideals of U n and ad-nilpotent ideals of b. (The exponential map?) The above theorem has been generalized from type A to other types [Krattenthaler Orsina Papi 2002]. Is there a similar generalization for nilpotent ideals of U n? Jia Huang (UNK) Combinatorics of non-associative binary operations December 26, / 28

25 Generating function Definition Let C d (x) := n 0 C d n x n+1 for d 1, and let C 0 (x) := x. Let F i (x) := i for i = 0, 1, and F n (x) := F n 1 (x) xf n 2 (x), n 2. Proposition (de Bruijn Knuth Rice 1972) For n 1 we have F n (x) = 0 i (n 1)/2 Proposition (Kreweras 1970) For d 1 we have C d (x) = ( n 1 i i ) ( x) i. x 1 C d 1 (x) = xf d+1(x) F d+2 (x). Example C 1 (x) = x 1 x, C 2 (x) = x 1 x 1 x = x(1 x) 1 2x, C 3 (x) = x 1 x 1 1 x x = x(1 2x) 1 3x+x 2 Jia Huang (UNK) Combinatorics of non-associative binary operations December 26, / 28

26 Closed Formulas for C d n Theorem (Andrews Krattenthaler Orsina Papi 2002) The number Cn d has the following closed formulas: Cn d = ( ) [( 2i(d + 2) + 1 2n + 1 i j + d = det 2n + 1 n i(d + 2) j i + 1 i Z ( ) ij+2 i j 1 =. i j+1 i j 0=i 0 i 1 i d 1 i d =n 0 j d 2 Theorem (de Bruijn Knuth Rice 1972) )] n 1 The number of plane trees with n + 1 nodes of depth at most d is C d n = 22n+1 d j d+1 sin 2 (jπ/(d + 2)) cos 2n (jπ/(d + 2)). i,j=1 Jia Huang (UNK) Combinatorics of non-associative binary operations December 26, / 28

27 More on the number C d n Theorem (Hein and H.) For n, d 1 we have C d,1 1,1,n = C d n. Definition A composition of n is a sequence α = (α 1,..., α l ) of positive integers such that α α l = n. Let max(α) := max{α 1,..., α l } and l(α) = l. Proposition (Hein and H.) For n, d 1 we have C d n = α =n max(α) (d+1)/2 ( ) d ( 1) n l(α) α1 α i l(α) ( ) d + 1 αi α i Jia Huang (UNK) Combinatorics of non-associative binary operations December 26, / 28

28 Thank you! Jia Huang (UNK) Combinatorics of non-associative binary operations December 26, / 28