[Paper Review] Combinatorics on permutation tableaux of type $A$ and type $B$

Sylvie Corteel, Jang Soo Kim|arXiv (Cornell University)|Jun 18, 2010

Advanced Combinatorial Mathematics11 references4 citations

TL;DR

This paper provides two bijective proofs of a generating function for unrestricted columns in permutation tableaux of type A, leading to a sign-imbalance formula when specialized to $ t = -1 $. It extends the Corteel-Nadeau bijection to type B permutation tableaux, generalizing a result of Lam and Williams and expressing both bijections as zigzag maps on alternative representations, thereby unifying combinatorial structures across types A and B.

ABSTRACT

We give another bijective proof of a result of Corteel and Nadeau. We find a generating function related to unrestricted columns of permutation tableaux. As a consequence, we obtain a sign-imbalance formula for permutation tableaux. We extend the first bijection of Corteel and Nadeau between permutations and permutation tableaux to type $B$ objects. Using this type $B$ bijection, we generalize a result of Lam and Williams. We prove that the bijection of Corteel and Nadeau and our type $B$ bijection can be expressed as zigzag maps on the alternative representation.

Motivation & Objective

• To provide two bijective proofs of a generating function for unrestricted columns in type A permutation tableaux.
• To derive a sign-imbalance formula for permutation tableaux by specializing the generating function at $ t = -1 $.
• To extend the Corteel-Nadeau bijection from type A to type B permutation tableaux.
• To generalize a result of Lam and Williams on type B permutation tableaux using the new bijection.
• To express both the original and type B bijections as zigzag maps on the alternative representation of permutation tableaux.

Proposed method

• Use of a generating function $ \sum_{n \geq 0} \sum_{T \in \mathcal{PT}(n)} t^{\operatorname{urc}(T)} x^n = \frac{1 + E_t(x)}{1 + (t-1)x E_t(x)} $, where $ E_t(x) = \sum_{n \geq 1} n(t)_{n-1} x^n $, to encode unrestricted columns.
• Construction of a bijection $ \Phi_{\mathrm{B}} $ between type B permutation tableaux and type B permutations, extending the Corteel-Nadeau bijection.
• Definition of the alternative representation of permutation tableaux via $ \uparrow $, $ \leftarrow $, and removal of $ 0 $'s and $ 1 $'s.
• Introduction of the zigzag map $ \zeta^{\rm alt}_{\mathrm{B}} $ on the alternative representation, which maps tableaux to signed permutations.
• Proof that $ \zeta^{\rm alt}_{\mathrm{B}} = \varphi \circ \Phi_{\mathrm{B}} $, where $ \varphi $ is defined via RL-minima and cycle decomposition.
• Use of recurrence relations and combinatorial arguments to verify the generating function and its special cases.

Experimental results

Research questions

• RQ1 How can the generating function for the number of unrestricted columns in type A permutation tableaux be bijectively proven?
• RQ2 What is the combinatorial interpretation of the sign-imbalance formula $ \sum_{T \in \mathcal{PT}(n)} \operatorname{sign}(T) = \frac{(1+i)^n + (1-i)^n}{2} $?
• RQ3 How can the Corteel-Nadeau bijection between permutations and type A permutation tableaux be extended to type B objects?
• RQ4 Can the generalized generating function $ \sum_{T \in \mathcal{PT}_{\mathrm{B}}(n)} x^{\operatorname{urr}(T)-1} y^{\operatorname{top}_{0,1}(T)} z^{\operatorname{diag}(T)} = (1+z)^n (x+y)_{n-1} $ be derived via a bijective method?
• RQ5 How do zigzag maps on the alternative representation relate to the Corteel-Nadeau and type B bijections?

Key findings

• The generating function for unrestricted columns in type A permutation tableaux is $ \sum_{n \geq 0} \sum_{T \in \mathcal{PT}(n)} t^{\operatorname{urc}(T)} x^n = \frac{1 + E_t(x)}{1 + (t-1)x E_t(x)} $, where $ E_t(x) = \sum_{n \geq 1} n(t)_{n-1} x^n $.
• When $ t = 2 $, the sum $ \sum_{T \in \mathcal{PT}(n)} 2^{\operatorname{urc}(T)} $ equals the number of connected permutations of $ \{1,2,\dots,n+1\} $, providing a combinatorial interpretation.
• For $ t = -1 $, the sign-imbalance formula yields $ \sum_{T \in \mathcal{PT}(n)} \operatorname{sign}(T) = \frac{(1+i)^n + (1-i)^n}{2} $, which evaluates to $ (-1)^k \cdot 2^{2k} $ if $ n \equiv 0,1 \pmod{4} $, 0 if $ n \equiv 2 \pmod{4} $, and $ (-1)^{k+1} \cdot 2^{2k+1} $ if $ n \equiv 3 \pmod{4} $.
• The bijection $ \Phi_{\mathrm{B}} $ extends the Corteel-Nadeau bijection to type B permutation tableaux and satisfies $ \sum_{T \in \mathcal{PT}_{\mathrm{B}}(n)} x^{\operatorname{urr}(T)-1} y^{\operatorname{top}_{0,1}(T)} z^{\operatorname{diag}(T)} = (1+z)^n (x+y)_{n-1} $, generalizing Lam and Williams' result.
• The zigzag map $ \zeta^{\rm alt}_{\mathrm{B}} $ on the alternative representation of type B permutation tableaux coincides with $ \varphi \circ \Phi_{\mathrm{B}} $, where $ \varphi $ is defined via RL-minima and cycle decomposition.
• The absolute values of the sign-imbalance for type A permutation tableaux and standard Young tableaux are equal to $ 2^{\lfloor n/2 \rfloor} $ when $ n \not\equiv 2 \pmod{4} $, suggesting a deeper combinatorial connection.