Symmetry of ascent and descent distributions on rectangular and staircase tableaux

TL;DR

The paper provides direct bijective proofs of symmetry between the distributions of ascents and descents for standard Young tableaux of rectangular shapes $\lambda=(n^k)$ and truncated staircase shapes $\lambda=(n,n-1,\dots,n-k+1)$. Central to the approach are arrow encodings that yield involutions $\varphi$, $\psi$, and $\beta$, establishing $\mathrm{des}(T)+\mathrm{des}(\varphi(T))=(k-1)(n+1)$ and $\mathrm{asc}(T)+\mathrm{asc}(\psi(T))=(k-1)(n-1)$, thereby proving Narayana-type symmetries $N(k,n,h)=N(k,n,(k-1)(n-1)-h)$. The methodology recovers the Lalanne--Kreweras involution in the two-row case, extends Sulanke’s conjectures to three rows, and introduces refined statistics via the bounce matrix $B(T)$. Additionally, the work situates these bijections within the broader setting of graded posets, descent polynomials, and poset-rowmotion concepts, and connects to canon permutations through Narayana-type statistics. The results offer a bijective, structurally transparent lens on descent/ascend symmetry and open avenues for rowmotion and generalized poset symmetry in more shapes.

Abstract

We give direct bijective proofs of the symmetry of the distributions of the number of ascents and descents over standard Young tableaux of shape $λ$, where $λ$ is a rectangle $(n,n,\dots,n)$ or a truncated staircase $(n,n-1,\dots,n-k+1)$. These can be viewed as instances of the more general symmetry of the distribution of descents over linear extensions of graded posets, for which previous proofs by Stanley and Farley were based on the theory of $P$-partitions and the involution principle, respectively. In the case of two-row rectangles $(n,n)$, our bijection is equivalent to the Lalanne--Kreweras involution on Dyck paths, which bijectively proves the symmetry of the Narayana numbers. Our bijections are defined in terms of certain arrow encodings of standard Young tableaux. This setup allows us to construct other statistic-preserving involutions on tableaux of rectangular shape, providing a simple proof of the fact that ascents and descents are equidistributed up to a shift, and proving a conjecture of Sulanke about certain statistics in the case of three rows. Finally, we use our bijections to define a possible notion of rowmotion on standard Young tableaux of rectangular shape, and to give a bijective proof of the symmetry of the number of descents on canon permutations, which have been recently studied as a variation of Stirling and quasi-Stirling permutations.

Figures (9)

• Figure 1: The arrow encodings of two standard Young tableaux of rectangular and staircase shapes.
• Figure 2: An example of the involution $\varphi_r$ on $\mathop{\mathrm{SYT}}\nolimits(6^5)$ for $r=3$. Each leading ${\bm\downarrow}$ in row $r$ and each trailing ${\bm\uparrow}$ in row $r+1$ is colored in red. In columns with a pair of red arrows $\textcolor{red}{\bm\downarrow}\textcolor{red}{\bm\uparrow}$, the map $\varphi_r$ removes them, and in columns with no red arrows, the map adds such a pair.
• Figure 3: Examples of the involutions $\varphi$, $\beta$, and $\psi$. In the top tableaux, all the leading ${\bm\downarrow}$ in row $r$ and the trailing ${\bm\uparrow}$ in row $r+1$ have the same color for each fixed $r$, to help visualize $\varphi$. After applying $\beta$, these become trailing ${\bm\downarrow}$ and leading ${\bm\uparrow}$, respectively.
• Figure 4: The definitions of $\varphi_r$ and $\psi_r$ fail to produce valid arrow arrays for the shape $\lambda=(2,1,1)$.
• Figure 5: An example of the involution $\psi$ on a tableau of staircase shape $\lambda=(5,4,3,2,1)$. The trailing ${\bm\downarrow}$ in row $r$ and the leading ${\bm\uparrow}$ in row $r+1$ have the same color for each fixed $r$.

Theorems & Definitions (61)

• Theorem 1.1: sulanke_generalizing_2004
• Theorem 1.2: sulanke_generalizing_2004
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Definition 3.1
• Definition 3.2
• Lemma 3.3
• proof