Partial Motzkin paths with air pockets of the first kind avoiding peaks, valleys or double rises

Let $P$ be a length $n$ PMAP ending at height $k\geqslant 0$ and avoiding the occurrences of $UD_{i}$ for $i\geqslant 1$. If the last step of $P$ is $U$, then $k\geqslant 1$ and we have $P=QU$, where $Q$ is a length $(n-1)$ MAP ending at height $k-1$ and avoiding the peaks ($Q$ can be the empty path). So, we obtain the first relation $f_{k}=zf_{k-1}+zg_{k-1}+zh_{k-1}$ for $k\geqslant 1$, anchored with $f_{0}=1$ by considering the empty path. If the last step of $P$ is a down-step $D_{i}$, $i\geqslant 1$, then we have $P=QD_{i}$, where $Q$ is a length $(n-1)$ PMAP ending at height $\ell\geqslant k+1$ with no up- and down-steps at its end, and with no peaks. So, we obtain the second relation $g_{k}=z\sum\limits_{\ell\geqslant k+1}h_{\ell}$. If the last step of $P$ is a horizontal step $H$, then we have $P=QH$, where $Q$ is a length $(n-1)$ PMAP ending at height $k$ with no peaks, which implies that $h_{k}=z(f_{k}+g_{k}+h_{k})$ for $k\geqslant 0$.

Therefore, we have to solve the following system of equations:

Multiplying by $u^{k}$ the recursions in (1) and summing over $k$, we have:

Notice that we have $F(1)-H(1)=1$ by considering the difference of the first and third equations. Now, setting $h1:=H(1)$ and solving these functional equations, we obtain

In order to compute $h1$, we use the kernel method (see [1, 11]) on $H(u)$. We can write the denominator (which is a polynomial in $u$ of degree 2), as $z(u-r)(u-s)$ with

Plugging $u=s$ (which has a Taylor expansion at $z=0$) in $H(u)z(u-r)(u-s)$, we obtain the equation $zh1-s+1=0,$ which implies that

Finally, after simplifying by the factor $(u-s)$ in the numerators and denominators, we obtain

which induces that

The first few terms of the series expansion of $\mathit{Total}(z,1)$ are

which corresponds to the sequence A152225 in [18] counting Dyck paths of semilength $n+1$ with no peaks of height $0$ (mod 3) and no valleys of height 2 (mod 3); see [7].

The first few terms of the series expansion of $\mathit{Total}(z,0)$ are

which corresponds to the sequence A152171 in [18] counting Dyck paths of length $2n$ with no peaks of height $2$ (mod 3) and no valleys of height 1 (mod 3). In Section 2.2, we will exhibit a constructive bijection between these two classes of paths.

Let $\mathcal{T}$ be the infinite matrix $\mathcal{T}:=[t(n,k)]_{n,k\geqslant 0}$, where $t(n,k)=[z^{n}][u^{k}]\mathit{Total}(z,u)$. The first few rows of the matrix $\mathcal{T}$ are

In Figure 1 we show the peak-less PMAP counted by $t(5,1)=15$.

In [15], Rogers gave an equivalent characterization of the Riordan arrays. That is, every element not belonging to row $0$ or column $0$ in a Riordan array can be expressed as a fixed linear combination of the elements in the preceding row. The $A$-sequence is defined to be the sequence coefficients of this linear combination. Analogously, Merlini et al. [9] introduced the $Z$-sequence, that characterizes the elements in column $0$, except for the top one.

An infinite lower triangular matrix $[d_{n,k}]_{n,k\geqslant 0}$ is a Riordan array if and only if $d_{0,0}\neq 0$ and there exist two sequences $(a_{0},a_{1},a_{2},\dots)$, with $a_{0}\neq 0$, and $(z_{0},z_{1},z_{2},\dots)$ (called the $A$-sequence and the $Z$-sequence, respectively), such that

The product of two Riordan arrays $(g(z),f(z))$ and $(h(z),l(z))$ is defined by

Under this operation, the set of all Riordan arrays is a group [16]. The identity element is $I=(1,z)$ and the inverse of $(g(z),f(z))$ is given by

where $f^{<-1>}(z)$ denotes the compositional inverse of $f(z)$.

The generating functions for the $A$-sequence and $Z$-sequence of the Riordan array $\mathcal{F}=(g(z),f(z))$, with inverse $\mathcal{F}^{-1}=(d(z),h(z))$, are given by ([9, 5])

respectively.

From the definition of the $A$-sequence and $Z$-sequence for the Riordan arrays we can give an additional recurrence relation for the sequence $t(n,k)$.

The first few values of the sequence $a(n)$ for $n\geqslant 0$ are

Corollary 2 proves that the set of peak-less Motzkin paths with air pockets of length $n$ (ending on the $x$-axis) is equinumerous to the set $\mathcal{D}_{n}(2,1)$ of Dyck paths of length $2n$ with no peak at height 2 (mod 3) and no valley at height 1 (mod 3).

Any non-empty peak-less Motzkin path with air pockets is either of the form (1) $H\alpha$ or (2) $U\alpha_{1}U\alpha_{2}\cdots U\alpha_{k}HD_{k}\beta$, where $k\geqslant 1$ and $\alpha,\alpha_{1},\ldots,\alpha_{k},\beta$ are possibly empty peak-less MAP. We refer to the left part of Figure 2 for an illustration of this form.

Remember that $\mathcal{MP}$ denotes the set of Motzkin paths with air pockets of the first kind.

We refer to Figure 3 for an illustration of the third case of the definition of $\psi$.

Due to the recursive definition, the image of peak-less MAP of length $n$ under $\psi$ is a Dyck path of length $2n$. Moreover it is clear that the obtained path is a Dyck path in $\cup_{n\geqslant 0}\mathcal{D}_{n}(2,1)$. For instance (see Figure 3 for an illustration of this example).

In the same way as we done in Section 2.1, we have to solve the following system of equations:

Multiplying by $u^{k}$ the recursions in (7) and summing over $k$, we have:

Notice that we have $(1-z)F(1)=1+zH(1)$ by considering the first equation. Now, setting $f1:=F(1)$ and solving these functional equations, we obtain

In order to compute $f1$, we use the kernel method (see [1, 11]) on $F(u)$. We can write the denominator (which is a polynomial in $u$ of degree 2), as $z(u-r)(u-s)$ with

Plugging $u=s$ (which has a Taylor expansion at $z=0$) in $F(u)z(u-r)(u-s)$, we obtain the equation ${\it f1}\,s{z}^{2}-s{z}^{2}+zs+{z}^{2}-s-z+1=0,$ which implies that

Finally, after simplifying by the factor $(u-s)$ in the numerators and denominators, we obtain

which implies that