Generalized Small Schröder Numbers

We study generalized small Schröder paths in the sense of arbitrary sizes of steps. A generalized small Schröder path is a generalized lattice path from (0, 0) to (2n, 0) with the step set of { set of positive integers, which never goes below the x-axis, and with no horizontal steps at level 0. We find a bijection between 5-colored Dyck paths and generalized small Schröder paths, proving that the number of generalized small Schröder paths is equal to n k=1 N (n, k)5 n−k for n 1.


Introduction
Lattice paths have been studied extensively by many mathematicians over a long period of time.Recently, several attempts of generalization have been successful.(See [2], [4], [7], and [3].)Schröder paths are also interesting objects, but have been rarely studied for any type of generalization until J.P.S Kung and A. De Mier [5] found the generating functions for the number of generalized Dyck and Schröder paths, which are called rook and queen paths, with a given right boundary and steps satisfying a natural slope condition.In this paper we try to enumerate combinatorially the number of the generalized small Schröder paths.
In the next section we give some basic definitions and facts on lattice paths in general.Section 3 talks about a bijection between colored Dyck paths and generalized small Schröder paths.

Preliminaries
We list two sets of definitions.
(2) An n-Dyck path is called a prime if it never touches the x-axis except at the starting and terminal vertices.
(3) A peak in a Dyck path is a vertex between an up step and a down step, and a valley is a vertex between a down step and an up step.
(4) A double ascent in a Dyck path is an intermediate vertex of a consecutive U U pair, and a double descent is defined similarly.
(5) The Narayana number N (n, k) counts n-Dyck paths with k peaks.
(7) An n-small Schröder path is a Schröder path having no flat steps at level 0.
Definition 2. We introduce the new ones that we need throughout this paper.
(1) A 5-colored n-Dyck path is an n-Dyck path only whose double ascents are colored in one of five colors, b(black), w(white), f (f lame), g(green), and y(yellow).
(2) A bicolored n-small Schröder path is an n-small Schröder path whose vertices between distinct steps of U, D, and F are colored in b only and vertices between the same kind of steps of U U, DD, and F F are colored in either b or w.The starting vertex and ending vertex are not colored.
(4) A generalized n-small Schröder path is a generalized n-Schröder path with no flat steps at level 0.

A bijection between 5-colored Dyck paths and generalized small Schröder paths
In this section we provide a bijective relation between 5-colored n-Dyck paths and bicolored n-small Schröder paths by Bijection I. Then we give a natural bijection between bicolored n-small Schröder paths and generalized n-small Schröder paths by Bijection II.
In other words, we prove the following theorem bijectively.
Theorem 4. The number of generalized n-small Schröder paths is Proof.Bijection I.
Let P be a 5-colored n-Dyck path.We construct the corresponding bicolored n-small Schröder paths by taking four steps described as follow: For each double ascent of P find the first double descent right of the ascent at the same level.We call this the matching double descent.Note that each double ascent of P is colored in one of five colors, b, w, f, g, and y.Let P be the path obtained from P by the following process: (i) If a double ascent of P is colored in one of three colors, b, w, and f , then color its matching double descent in the same color.
(ii) If a double ascent of P is colored in one of two colors, g and y, then replace the color with b and w, respectively, and color its matching double descent in w and b, respectively.(iii) Color each peak and valley of P in b.
Notice that every f colored double ascent in P has the matching double descent also colored in f .

STEP 2. (factorizing)
Find the first vertex v colored in f .Assume that v is the ending vertex of an up step U at level i.Then one can find the corresponding first down step D at the same level.Let Q be the subpath of P above the level i from U to D. Then we can factor P in general as LU f QxDR, where L and R are subpaths of P , f is the color of the vertex v, and x is a color of the starting vertex of D which is either b or f .Initially, this x is f , but as processes repeated it can be b if Q ends with a flat step.(See Figure 1 (i).)

STEP 3. (creating flats)
In the subpath U f QxD, Q can be factored as Now find the lowest level k of Q 1 at which a black colored vertex lies.
the electronic journal of combinatorics 22(3) (2015), #P3.14Note that (i) Q 1 starts with k − i U steps and ends with k − i D steps.(ii) First k − i − 1 double ascents are colored in w or f (iii) Each of the last k − i − 1 double descents is colored in the same color of its matching double ascent.

Thus we can factor
1 is a subpath of Q 1 above the level k, U j is the up step ending at the level i + j, and D j is the down step starting at the level i We explain these four cases one at a time, all of which will be concluded by the last process of changing colors.We start at the subpath U f QxD of the path P .Then the path U f QxD is now transformed to

Case 1. (i) Erase the color
and then flat steps followed.
The first three processes are the same as Case 1.We just change the color of the first flat step from (ii) Replace each up step U j (1 (iv) Cut off bQ 2 of Q = Q 1 bQ 2 and then paste it after turning 180 degrees (denoted by Q2 b) in between the first color b in (i) and the first flat step F 1 in (ii).(See Figure 4).(i) Erase the color f of U f QxD and replace the x with b.

Then the path
(iv) Let Q 2 represent the newly created flat steps by (iii) along with Q 2 b from (i), and after turning 180 degrees of Q 2 let us denote it by Q 2 .Now cut off Q 2 and then paste Q the electronic journal of combinatorics 22(3) (2015), #P3.14For the last process of Step 3 we replace the color f 's between the flat steps with b's.STEP 4. Repeat Step 2 and 3 until there is no vertex colored in f .Then clearly the outcome is a bicolored n-small Schröder path.Now we give an example, which can be obtained by applying STEPs 1 through 4 along with Figures 1 through 5.
Let a 5-colored 16-Dyck path P be given: where D 5 represents DDDDD.We will construct the corresponding bicolored 16-small Schröder path P : P = U wU bF bU bF bU bDwDbF wF bF bU bF bDbF bF bU bDbF bU bDwD.
Step by step applications of STEPs and Cases are showed in the Figure 6.
For recovery processes we first note that from STEP 3: the electronic journal of combinatorics 22(3) (2015), #P3.14 2. Cases 3 and 4 produce the color w to the starting vertex of the D in U f QxD, and the rightmost block of F 's on the level i is created lastly.
3. In any case, if Q 2 is not empty, then it should not start with a flat step.Now we describe the recovery process.Let P be a bicolored n-small Schröder path.First, we label the ith up step by U i from left to right.Let U i be an up step ending at the level k and let U j (i < j) be the next up step ending at the level k.Then the flat steps at the level k between these up steps U i and U j are labeled as F i .If there is no such U j , then every flat step at level k is labeled as F i .We call this U i a label up step for the flat steps.We also call the first down step after U i starting at the same level the label down step for the flat steps.Now find the flat steps whose subscript is the largest.These steps can be appeared as blocks of consecutive flat steps.Then look for the starting vertex v 2 of the label down step for these flat steps.Let v 1 be the ending vertex of the corresponding label up step, for convenience.Then we take the following processes to recover the corresponding Dyck path just before the STEP 1.
1.If v 2 has color b, then the leftmost block of flat steps are changed first.If the color of v 1 is b or w, then we apply the reverse process of Case 1 of STEP 3. If the color of v 1 is f , then apply the reverse process of Case 2 of STEP 3.
2. If v 2 has the color w, then the rightmost block of flat steps are changed first.If the color of v 1 is b, then we apply the reverse process of Case 3 of STEP 3. If the color of v 1 is w, then apply the reverse process of Case 4 of STEP.
3. Repeat these processes until we have no flats.
Then we have an n-Dyck path whose double ascents and double descents are colored in one of three colors, b, w, and f , and all peaks and valleys are colored in b.By the following recovery processes of colors, we can fully recover the 5-colored n-Dyck path P : 1. Delete the colors of peaks and valleys.
2. If a double ascent and its matching double descent are in the same color, then delete the color of the double descent.

If not, give each double ascent a new color by the following:
(a) Color the double ascent in g if a pair of the colors of a double ascent and its matching double descent is (b, w) and then delete the color of the double descent.
(b) Color the double ascent in y if a pair of the colors of a double ascent and its matching double descent is (w, b) and then delete the color of the double descent.
Now we have the 5-colored n-Dyck path as we desired.Figure 7 shows full recovery processes.
Let us consider the second bijection.As we mentioned earlier, this bijection gives oneto-one correspondence between bicolored n-small Schröder paths and generalized n-small Schröder paths.

Bijection II.
Let P be a given bicolored n-small Schröder path.Recall that in P all peaks and valleys are colored in black, and the starting and ending vertices of all blocks of flats are colored in black.And other vertices can be black within blocks of up, down, and flat

Figure 1 :
Figure 1: The factorizations of P , (i) and Q, (ii).A square vertex represents a vertex of color f .
We have four cases to construct flat steps depending on the colors of vertices c k−i and c k−i+1 , i.e., the starting and ending vertices of the subpath Q * 1 , which are: Case 1: The color c k−i+1 is b.Case 2: The colors of c k−i and c k−i+1 are both f .Case 3: The color c k−i is b and c k−i+1 is w.Case 4: The colors of c k−i and c k−i+1 are both w.
so that the color c k−i takes over the color f .(ii) Replace each down step D j (1 j k − i) of Q 1 by F j at level i with colors unchanged.(iii) Replace the x of U f QxD by b if it is an f .the electronic journal of combinatorics 22(3) (2015), #P3.14