On Promotion and Quasi-Tangled Labelings of Posets

Let $P$ be an $n$-element poset, whose order relation we denote by $<_{P}$. A labeling of $P$ is a bijection $L:P\to[n]$ (where $[n]=\{1,\ldots,n\}$). A labeling $L$ is called a linear extension if it preserves the order on $P$, i.e. if for all pairs $x,y\in P$ with $x<_{P}y$ we have $L(x)<L(y)$. Let $\Lambda(P)$ be the set of all labelings of $P$; let $\mathcal{L}(P)\subset\Lambda(P)$ be the subset consisting of all linear extensions.

In [Sch63, Sch72, Sch76], Schützenberger introduced an intriguing bijection on $\mathcal{L}(P)$ called promotion. Promotion has connections with various topics in algebraic combinatorics and representation theory, as seen in [EG87, Hua20, PPR09, Rho10, Sta09].

In [DK22], Defant and Kravitz extended the promotion map to an operator $\partial:\Lambda(P)\to\Lambda(P)$, not necessarily invertible, that is defined on all of $\Lambda(P)$. When the poset is a chain, extended promotion is dynamically equivalent to the bubble-sort map studied in [Knu98]. Promotion can also be described in terms of Bender-Knuth involutions (first introduced by Haiman [Hai92] as well as Malvenuto and Reutenauer [MR94]); in [DK22], Defant and Kravitz extended these Bender-Knuth involutions to arrive at an equivalent “toggle” definition of extended promotion. The following results of [DK22] are crucial properties of extended promotion:

1. (1)

   When restricted to $\mathcal{L}(P)$, $\partial$ agrees with Schützenberger’s promotion operator.

2. (2)

   If $L$ is a labeling of an $n$-element poset $P$, then $\partial^{n-1}(L)\in\mathcal{L}(P)$.

Thus, (extended) promotion¹¹1Henceforth, “promotion” always refers to $\partial$ rather than its restriction to $\mathcal{L}(P)$. may be regarded as a sorting operator, where linear extensions are considered “sorted.” Property (2) shows that promotion sorts every labeling after at most $n-1$ applications.

We define the sorting time of a labeling $L$ to be the smallest $k\in\mathbb{N}$ such that $\partial^{k}(L)\in\mathcal{L}(P)$. Defant and Kravitz mainly studied tangled labelings—those labelings with sorting time $n-1$. In particular, they enumerated these tangled labelings for a large class of posets called inflated rooted forests. They also studied sortable labelings—those labelings $L$ such that $\partial(L)\in\mathcal{L}(P)$—and enumerated these sortable labelings for arbitrary posets.

In Section 2, we present the main definitions and background results needed for the rest of the paper. In Section 3, we study the cardinality of $\partial^{-1}(L)$ for an arbitrary labeling $L$. Our 3.3 gives a formula for $|\partial^{-1}(L)|$ when $P$ is a rooted tree poset. We also study the degree of noninvertibility of promotion and give a sharp lower bound for this (3.7). Section 4 contains an explicit enumeration of the labelings with sorting time $n-2$ for a large class of posets called inflated rooted trees with deflated leaves (4.6). A corollary of this result is that an $n$-element poset with a unique minimal element with the property that the minimal element has exactly one parent has $2(n-1)!-(n-2)!$ quasi-tangled labelings. In Section 5, we present an algorithmic approach to enumerating the labelings of a rooted tree poset with sorting time $n-k-1$ for fixed $k\in\{1,\ldots,n-2\}$, and in Section 6, we make partial progress (6.4) on the following conjecture:

Let $P$ be an $n$-element poset, and let $L$ be a labeling of $P$. For $x\in P$ not maximal, the L-successor of $x$ is the element greater than $x$ with minimal label. Now, let $v_{1}=L^{-1}(1)$. Let $v_{2}$ be the $L$-successor of $v_{1}$; let $v_{3}$ the $L$-successor of $v_{2}$, and so on until we get an element $v_{m}$ that is maximal. The resulting chain $v_{1}<_{P}v_{2}<_{P}\cdots<_{P}v_{m}$ is called the promotion chain of $L$. Now, define $\partial(L)$ to be the labeling

In other words, promotion may be thought of as decreasing each label by 1 (working modulo $n$ so that $0=n$) and then cycling the promotion chain downwards one step. The following proposition captures a fundamental sorting property of promotion.

A lower order ideal of a poset $P$ is a subset $Q\subset P$ such that for every $x\in Q$ and $y\in P$ with $y<_{P}x$ we have $y\in Q$. Similarly, an upper order ideal of a poset $P$ is a subset $Q\subset P$ such that for all $x\in Q$ and $y\in P$ with $y>_{P}x$ we have $y\in Q$. It is often useful to note that $Q$ is a lower order ideal of $P$ if and only if $P\setminus Q$ is an upper order ideal of $P$. For $x,y\in P$, we say that $y$ covers $x$ and write $x\lessdot y$ if $x<_{P}y$ and $\{z\in P\;|\;x<_{P}z<_{P}y\}=\emptyset$. In this case, we say that $y$ is a parent of $x$ and that $x$ is a child of $y$.

The Hasse diagram of a poset $P$ is a graphical illustration of its covering relations. Each element of $P$ is represented by a vertex, and if $x<_{P}y$, then the vertex corresponding to $x$ is drawn below that corresponding to $y$; there exists an edge between these vertices if and only if $x\lessdot_{P}y$. We say a poset is connected if its Hasse diagram is connected when regarded as a graph; the connected components of $P$ are the subposets induced by the connected components of the Hasse diagram of $P$.

Suppose $P$ is an $n$-element poset, and let $f:P\to\mathbb{Z}$ be an injective function. Then the standardization of $f$, denoted $\mathrm{st}(f)$, is the labeling $L:P\to[n]$ such that for all $x,y\in P$, $L(x)<L(y)$ if and only if $f(x)<f(y)$. Note that this labeling is unique. Equivalently, if $g:f(P)\to[n]$ is an order-preserving bijection, then $\mathrm{st}(f)=g\circ f$.

1.2 motivates the following definitions:

We let the set of all tangled labelings of $P$ be denoted by $\mathcal{T}(P)$; we denote the set of all sortable labelings by $\Sigma(P)$. If $P$ is a poset, $L$ a labeling of $P$, and $\gamma\in\mathbb{N}$, we also will frequently use the shorthand $L_{\gamma}$ to denote $\partial^{\gamma}(L)$. Note that $L_{0}=L$.

Note that $L$ is a linear extension if and only if all elements of $P$ are frozen. Also, we remark that it is possible for a labeling of an $n$-element poset to have no frozen elements, namely, when $L^{-1}(n)$ is not maximal.

Defant and Kravitz proved the following useful results about frozen elements.

Note that the $\gamma=n-1$ case of Lemma 2.6 immediately implies 1.2.

The following results will also be helpful later:

Note here that a rooted tree poset is a poset whose Hasse diagram is a rooted tree where the root is the unique maximal element. A rooted forest poset is a poset whose connected components are rooted tree posets.

In the following, we give a formula enumerating the quasi-tangled labelings of inflated rooted tree posets with deflated leaves. The following is shown in the proof of 2.9, but we state it as its own lemma here:

4.6 enumerates the quasi-tangled labelings of inflated rooted trees with deflated leaves. In particular, note that this relatively large class of posets includes rooted tree posets. While this particular class of posets seems artificial, it will be important that the posets we are working with have the property that if one removes a minimal element or an element covering a minimal element from the poset, then the resulting poset remains an inflated rooted tree.

We remark here that the quasi-tangled labelings of a poset are those labelings $L$ such that $L_{n-3}$ is not a linear extension but $L_{n-2}$ is a linear extension, i.e., $L$ is not tangled. To count these labelings, we condition on the positions of $L^{-1}(n-1)$ and $L^{-1}(n)$ and keep track of where $L_{\gamma}^{-1}(n-1-\gamma)$ and $L_{\gamma}^{-1}(n-2-\gamma)$ end up (after $n-3$ promotions these are the elements that are labeled 2 and 1, respectively). In particular, we consider a uniformly random labeling $L$ of $P$ with $L^{-1}(n)$ or $L^{-1}(n-1)$ fixed and calculate the probability that, at each “branch vertex,” the elements in question “get pulled down” in the desired direction, i.e., in the direction such that $L_{n-3}$ is not a linear extension. The proof of the theorem involves a lot of casework, which we prove beforehand in several technical lemmas.

Let $(P,\varphi)$ be an inflation of a rooted tree poset $Q$. For each leaf $\ell$ of $Q$, there exists a unique path in the Hasse diagram of $Q$ from $\ell$ to the root. Let the elements of this path be called $u_{\ell,0},u_{\ell,1},\ldots,u_{\ell,\omega(\ell)}$, where $u_{\ell,0}=\ell$ and for all $1\leq j\leq\omega(\ell)$, $u_{\ell,j}$ covers $u_{\ell,j-1}$. Then define

Note that $\frac{b_{\ell,j}}{c_{\ell,j}}$ is the fraction of elements below the minimal element of $\varphi^{-1}(u_{\ell,j})$ that “lie in the direction” of $\varphi^{-1}(\ell)$. The following enumerates the number of tangled labelings of $P$:

Now, define $M$ to be the set of minimal elements of $P$. We define subsets $R,S,T$ of $M$: Let $\ell\in M$, and let $x$ cover $\ell$.

1. (1)

   Put $\ell$ in $R$ if $\ell$ is the only child of $x$ and the parent of $x$ exists and has multiple children.

2. (2)

   Put $\ell$ in $S$ if $x$ has precisely two children;

3. (3)

   Put $\ell$ in $T$ if $\ell$ is the only child of $x$, the parent of $x$ exists, and $x$ is the only child of its parent.

Note here that $R$, $S$, and $T$ are disjoint but do not partition $P$. However, it is not difficult to see that $R$, $S$, and $T$ partition the set of minimal elements of $P$ that, along with their parents, lie in non-antichain lower order ideals of size 3. See Figure 3.

In this section, we compute several probabilities that will help us in the proof of 4.6. Importantly, in Lemma 4.11, we generalize Lemma 3.11 of [DK22], which tells us that the probability of a certain label ending up in some subtree of our inflated rooted tree poset after a certain number of promotions is proportional to the size of the subtree.

For the rest of this subsection, let $P$, $x$, and $X$ be defined as in Lemma 4.8. Suppose there is a partition of $X$ into disjoint subsets $A$ and $B$ such that no element of $A$ is comparable to an element of $B$. Note that both $A$ and $B$ are lower order ideals of $P$.

With notation as above, note that in order to determine whether $L_{k}^{-1}(m)\in A$, it suffices to determine whether $L_{\gamma}^{-1}(1)\in A$: because $L_{\gamma}^{-1}(m+k-\gamma)\not\in X$ and $L_{\gamma+1}^{-1}(m+k-\gamma-1)\in X$, we have that $L_{\gamma}^{-1}(m+k-\gamma)$ is in the $\gamma$th promotion chain. Hence, $L_{\gamma+1}^{-1}(m+k-\gamma-1)\in A$ if and only if $L_{\gamma}^{-1}(1)\in A$ (recall $A$ and $B$ are disjoint lower order ideals).

To each such $m+k$, one can associate a decreasing sequence of indices $\gamma_{0},\gamma_{1},\ldots,\gamma_{r}$ whose values depend only on $L|_{P\setminus X}$ as follows. Let $\gamma_{0}$ pull down $m+k$. By Lemma 4.9, the value $\gamma_{0}$ depends only on $L|_{P\setminus X}$. If $L^{-1}(\gamma_{0}+1)\in X$, we are done. Otherwise, let $\gamma_{1}$ pull down $\gamma_{0}+1$, where we let $\gamma_{0}$ take the role of $k$ and $1$ the role of $m$ (note that $\gamma_{1}<\gamma_{0}$). If $L^{-1}(\gamma_{1}+1)\in X$, we are done; otherwise, let $\gamma_{2}$ pull down $\gamma_{1}+1$. This process can be continued, where $\gamma_{i+1}$ pulls down $\gamma_{i}+1$. Since $0\leq\gamma_{i+1}<\gamma_{i}$, this process eventually terminates, yielding a decreasing sequence $\gamma_{0},\ldots,\gamma_{r}$. By Lemma 4.9, we see that the values $\gamma_{0},\ldots,\gamma_{r}$ depend only on the set $L(X)$ and $L|_{P\setminus X}$, not on the way in which the labels in $L(X)$ are distributed.

Let $a,b\in[N]$ and $k\in[N-1]$ be such that $a+k,b+k\leq N$, $L_{k}^{-1}(a),L_{k}^{-1}(b)\in X$, and $L^{-1}(a+k),L^{-1}(b+k)\not\in X$. Suppose $a\neq b$. Let $\alpha_{0},\ldots,\alpha_{r}$ and $\beta_{0},\ldots,\beta_{s}$ be the sequences associated to $a+k$ and $b+k$, respectively. We claim that $\{\alpha_{0},\ldots,\alpha_{r}\}\cap\{\beta_{0},\ldots,\beta_{s}\}=\emptyset$. Assume the contrary. If $\alpha_{i}=\beta_{j}$ for some $0\leq i\leq r$ and $0\leq j\leq s$, then it follows from 4.10 that $\alpha_{r}=\beta_{s}$.

Without loss of generality, suppose $r\leq s$. If $r<s$, then by definition, $\alpha_{r}=\beta_{s}$ pulls down $\alpha_{r-1}+1=\beta_{s-1}+1$, $\alpha_{r-1}=\beta_{s-1}$ pulls down $\alpha_{r-2}+1=\beta_{s-2}+1$, etc., until $\alpha_{0}=\beta_{s-r}$ pulls down $a+k=\beta_{s-r-1}+1$. It follows that $a+k-1=\beta_{s-r-1}$. Note that $a+k-1\geq k$, but $\beta_{s-r-1}\leq\beta_{0}<k$. This is a contradiction. The case where $s>r$ is identical. If $r=s$, then $\alpha_{0}=\beta_{0}$, and it follows that $a+k=b+k$, contradicting our assumption that $a\neq b$.