Canon permutations and generalized descents of standard Young tableaux

Abstract

Canon permutations are permutations of the multiset having $k$ copies of each integer between $1$ and $n$, with the property that the subsequences obtained by taking the $j$th copy of each entry, for each fixed $j$, are all the same. For $k=2$, canon permutations are sometimes called nonnesting permutations, and it is known that the polynomial that enumerates them by the number of descents factors as a product of an Eulerian polynomial and a Narayana polynomial. We extend this result to arbitrary $k$, and we relate the problem to the enumeration of standard Young tableaux of rectangular shape with respect to generalized descent statistics. Our proof is bijective, and it also settles a conjecture of Sulanke about the distribution of certain lattice path statistics.

Figures (6)

• Figure 1: The nonnesting permutation $3532521414\in\mathcal{C}^2_5$ viewed as a labeled nonnesting matching.
• Figure 2: The bijection from Lemma \ref{['lem:sigmaT']}, with $\sigma=35142$. Note that $\mathop{\mathrm{Des}}\nolimits_\sigma(T)=\mathop{\mathrm{Des}}\nolimits(\pi)=\{2,4,7,9,10,11,14\}$.
• Figure 3: The bijections $f_{rs}$ and $F_{rs}$ for $r=1$ and $s=3$. The blocks $B,B',B"$ containing two different letters and underlined. Letting $\sigma=2431$ and $\tau=3421$, we have $\mathop{\mathrm{des}}\nolimits_\sigma(T)=10=\mathop{\mathrm{des}}\nolimits_\tau(f_{13}(T))$, $\mathop{\mathrm{plat}}\nolimits(T)=4=\mathop{\mathrm{plat}}\nolimits(f_{13}(T))$, and $\mathop{\mathrm{Des}}\nolimits_\sigma(T)=\{2,4,5,8,10,14,16,18,20,22\}=\mathop{\mathrm{Des}}\nolimits_\tau(F_{13}(T))$.
• Figure 4: The bijection $g_{\ell m}$ for $\ell=2$ and $m=5$. The blocks $B$ and $B'$ in case (2) of Definition \ref{['def:glm']} are underlined. Letting $S=\{2,5\}$, $\lambda=78456123$ and $\lambda'=78123456$, we have $\mathop{\mathrm{des}}\nolimits_\lambda(T)=9=\mathop{\mathrm{des}}\nolimits_{\lambda'}(g_{25}(T))+1$ and $\mathop{\mathrm{plat}}\nolimits(T)=\mathop{\mathrm{plat}}\nolimits(g_{25}(T))=0$.
• Figure 5: The bijection $g_S=g_{02}\circ g_{25}\circ g_{56}$ for $S=\{2,5,6\}$. Letting $\lambda=78456312$, we have $\mathop{\mathrm{des}}\nolimits_\lambda(T)=10=\mathop{\mathrm{des}}\nolimits_{12\dots 8}(g_S(T))+3$ and $\mathop{\mathrm{plat}}\nolimits(T)=\mathop{\mathrm{plat}}\nolimits(g_{S}(T))=0$.

Theorems & Definitions (16)

• Theorem 1.1: elizalde_descents_2023
• Proposition 2.1: sulanke_generalizing_2004
• Definition 2.2
• Lemma 2.3
• proof
• Theorem 3.1
• Definition 4.1
• Definition 4.2
• Lemma 4.3
• proof