
We launch a systematic study of the refined Wilf-equivalences by the statistics comp and iar, where comp(π) and iar(π) are the number of components and the length of the initial ascending run of a permutation π, respectively. As Comtet was the first one to consider the statistic comp in his book Analyse combinatoire, any statistic equidistributed with comp over a class of permutations is called by us a Comtet statistic over such class. This work is motivated by a triple equidistribution result of Rubey on 321-avoiding permutations, and a recent result of the first and third authors that iar is a Comtet statistic over separable permutations. Some highlights of our results are:
● Bijective proofs of the symmetry of the joint distribution (comp,iar) over several Catalan and Schröder classes, preserving the values of the left-to-right maxima.
● A complete classification of comp- and iar-Wilf-equivalences for length 3 patterns and pairs of length 3 patterns. Calculations of the (des,iar,comp) generating functions over these pattern avoiding classes and separable permutations.
● A further refinement of Wang's descent-double descent-Wilf equivalence between separable permutations and (2413,4213)-avoiding permutations by the Comtet statistic iar.
Citation: Shishuo Fu, Zhicong Lin, Yaling Wang. Refined Wilf-equivalences by Comtet statistics[J]. Electronic Research Archive, 2021, 29(5): 2877-2913. doi: 10.3934/era.2021018
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We launch a systematic study of the refined Wilf-equivalences by the statistics comp and iar, where comp(π) and iar(π) are the number of components and the length of the initial ascending run of a permutation π, respectively. As Comtet was the first one to consider the statistic comp in his book Analyse combinatoire, any statistic equidistributed with comp over a class of permutations is called by us a Comtet statistic over such class. This work is motivated by a triple equidistribution result of Rubey on 321-avoiding permutations, and a recent result of the first and third authors that iar is a Comtet statistic over separable permutations. Some highlights of our results are:
● Bijective proofs of the symmetry of the joint distribution (comp,iar) over several Catalan and Schröder classes, preserving the values of the left-to-right maxima.
● A complete classification of comp- and iar-Wilf-equivalences for length 3 patterns and pairs of length 3 patterns. Calculations of the (des,iar,comp) generating functions over these pattern avoiding classes and separable permutations.
● A further refinement of Wang's descent-double descent-Wilf equivalence between separable permutations and (2413,4213)-avoiding permutations by the Comtet statistic iar.
In this paper, we study the behavior of the solutions of the following three-dimensional system of difference equations of second order
xn+1=f(yn,yn−1),yn+1=g(zn,zn−1),zn+1=h(xn,xn−1) | (1) |
where
Now, we explain our motivation for doing this work. Clearly if we take
xn+1=f(yn,yn−1),yn+1=g(xn,xn−1). | (2) |
Noting also that if we choose
xn+1=f(xn,xn−1). | (3) |
In [23], the behavior of the solutions of System (2) has been investigated. System (2) is a generalization of Equation (3), studied in [17]. The present System (1) is the three-dimensional generalization of System (2).
In the literature there are many studies on difference equations defined by homogeneous functions, see for instance [1,2,5,7,11,16]. Noting that also a lot of studies are devoted to various models of difference equations and systems, not necessary defined by homogeneous functions, see for example [4,9,10,12,14,18,19,20,21,22,24,25,26,27,28,29].
Before we state our results, we recall the following definitions and results. For more details we refer to the following references [3,6,8,13].
Let
Yn+1=F(Yn),n∈N0, | (4) |
where the initial value
Definition 1.1. Let
limn→∞Yn=¯Y. |
limn→∞Yn=¯Y. |
Assume that
Zn+1=FJ(¯Y)Zn,n∈N0,Zn=Yn−¯Y, |
where
To study the stability of the equilibrium point
Theorem 1.2. Let
(i) If all the eigenvalues of the Jacobian matrix
(ii) If at least one eigenvalue of
Definition 1.3. A solution
xn+p=xn,yn+p=yn,zn+p=zn,n≥−1. | (5) |
The solution
Definition 1.4. Let
Remark 1. For every term
Definition 1.5. A function
Φ(λu,λv)=λmΦ(u,v) |
for all
Theorem 1.6. Let
1. Then,
u∂Φ∂u(u,v)+v∂Φ∂v(u,v)=mΦ(u,v),(u,v)∈(0,+∞)2. |
(This statement is usually called Euler's Theorem).
2. If
A point
¯x=f(¯y,¯y),¯y=g(¯z,¯z),¯z=g(¯x,¯x). |
Using the fact that
(¯x,¯y,¯z)=(f(1,1),g(1,1),h(1,1)) |
is the unique equilibrium point of System (1).
Let
F(W)=(f1(W),f2(W),g1(W),g2(W),h1(W),h2(W)),W=(u,v,w,t,r,s) |
with
f1(W)=f(w,t),f2(W)=u,g1(W)=g(r,s),g2(W)=w,h1(W)=h(u,v),g2(W)=r. |
Then, System (1) can be written as follows
Wn+1=F(Wn),Wn=(xn,xn−1,yn,yn−1,zn,zn−1)t,n∈N0. |
So,
¯W=(¯x,¯x,¯y,¯y,¯z,¯z)=(f(1,1),f(1,1),g(1,1),g(1,1),h(1,1),h(1,1)) |
is an equilibrium point of
Assume that the functions
Xn+1=JFXn,n∈N0 |
where
¯W=(f(1,1),f(1,1),g(1,1),g(1,1),h(1,1),h(1,1)). |
We have
JF=(00∂f∂w(¯y,¯y)∂f∂t(¯y,¯y)001000000000∂g∂r(¯z,¯z)∂g∂s(¯z,¯z)001000∂h∂u(¯x,¯x)∂h∂v(¯x,¯x)0000000010) |
As
¯y∂f∂w(¯y,¯y)+¯y∂f∂t(¯y,¯y)=0 |
which implies
∂f∂t(¯y,¯y)=−∂f∂w(¯y,¯y). |
Similarly we get
∂g∂s(¯z,¯z)=−∂g∂r(¯z,¯z),∂h∂v(¯x,¯x)=−∂h∂u(¯x,¯x). |
It follows that
JF=(00∂f∂w(¯y,¯y)−∂f∂w(¯y,¯y)001000000000∂g∂r(¯z,¯z)−∂g∂r(¯z,¯z)001000∂h∂u(¯x,¯x)−∂h∂u(¯x,¯x)0000000010) |
The characteristic polynomial of the matrix
P(λ)=λ6−∂h∂u(¯x,¯x)∂g∂r(¯z,¯z)∂f∂w(¯y,¯y)λ3+3∂h∂u(¯x,¯x)∂g∂r(¯z,¯z)∂f∂w(¯y,¯y)λ2−3∂h∂u(¯x,¯x)∂g∂r(¯z,¯z)∂f∂w(¯y,¯y)λ+∂h∂u(¯x,¯x)∂g∂r(¯z,¯z)∂f∂w(¯y,¯y). |
Now assume that
|∂h∂u(¯x,¯x)∂g∂r(¯z,¯z)∂f∂w(¯y,¯y)|<18 |
and consider the two functions
Φ(λ)=λ6, |
Ψ(λ)=−∂h∂u(¯x,¯x)∂g∂r(¯z,¯z)∂f∂w(¯y,¯y)λ3+3∂h∂u(¯x,¯x)∂g∂r(¯z,¯z)∂f∂w(¯y,¯y)λ2−3∂h∂u(¯x,¯x)∂g∂r(¯z,¯z)∂f∂w(¯y,¯y)λ+∂h∂u(¯x,¯x)∂g∂r(¯z,¯z)∂f∂w(¯y,¯y). |
We have
|Ψ(λ)|≤8|∂h∂u(¯x,¯x)∂g∂r(¯z,¯z)∂f∂w(¯y,¯y)|<1=|Φ(λ)|,∀λ∈C:|λ|=1. |
So, by Rouché's Theorem it follows that all roots of
Hence, by Theorem 1.2, we deduce from the above consideration that the equilibrium point
Using Part 2. of Theorem 1.6 and the fact that the function
∂f∂w(¯y,¯y)=∂f∂w(1,1)¯y,∂g∂r(¯z,¯z)=∂g∂r(1,1)¯z,∂h∂u(¯x,¯x)=∂h∂u(1,1)¯x. |
In summary, we have proved the following result.
Theorem 2.1. Assume that
(¯x,¯y,¯z)=(f(1,1),g(1,1),h(1,1)) |
of System (1) is locally asymptotically stable if
|∂f∂u(1,1)∂g∂u(1,1)∂h∂u(1,1)|<f(1,1)g(1,1)h(1,1)8. |
Now, we will prove some general convergence results. The obtained results allow us to deal with the global attractivity of the equilibrium point
Theorem 2.2. Consider System (1). Assume that the following statements are true:
1.
a≤f(u,v)≤b,α≤g(u,v)≤β,λ≤h(u,v)≤γ,∀(u,v)∈(0,+∞)2. |
2.
3.
m1=f(m2,M2),M1=f(M2,m2),m2=g(m3,M3),M2=g(M3,m3),m3=h(m1,M1),M3=h(M1,m1) |
then
m1=M1,m2=M2,m3=M3. |
Then every solution of System (1) converges to the unique equilibrium point
(¯x,¯y,¯z)=(f(1,1),g(1,1),h(1,1)). |
Proof. Let
m01:=a,M01:=b,m02:=α,M02:=β,m03:=λ,M03:=γ |
and for each
mi+11:=f(mi2,Mi2),Mi+11:=f(Mi2,mi2), |
mi+12:=g(mi3,Mi3),Mi+12:=g(Mi3,mi3), |
mi+13:=h(mi1,Mi1),Mi+13:=h(Mi1,mi1). |
We have
a≤f(α,β)≤f(β,α)≤b, |
α≤g(λ,γ)≤g(γ,λ)≤β, |
λ≤h(a,b)≤h(b,a)≤γ, |
and so,
m01=a≤f(m02,M02)≤f(M02,m02)≤b=M01, |
m02=α≤g(m03,M03)≤g(M03,m03)≤β=M02, |
and
m03=λ≤h(m01,M01)≤g(M01,m01)≤γ=M03. |
Hence,
m01≤m11≤M11≤M01, |
m02≤m12≤M12≤M02, |
and
m03≤m13≤M13≤M03. |
Now, we have
m11=f(m02,M02)≤f(m12,M12)=m21≤f(M12,m12)=M21≤f(M02,m02)=M11, |
m12=g(m03,M03)≤g(m13,M13)=m22≤g(M13,m13)=M22≤g(M03,m03)=M12, |
m13=h(m01,M01)≤h(m11,M11)=m23≤h(M11,m11)=M23≤h(M01,m01)=M13, |
and it follows that
m01≤m11≤m21≤M21≤M11≤M01, |
m02≤m12≤m22≤M22≤M12≤M02, |
and
m03≤m13≤m23≤M23≤M13≤M03. |
By induction, we get for
a=m01≤m11≤...≤mi−11≤mi1≤Mi1≤Mi−11≤...≤M11≤M01=b, |
α=m02≤m12≤...≤mi−12≤mi2≤Mi2≤Mi−12≤...≤M12≤M02=β, |
and
λ=m03≤m13≤...≤mi−13≤mi3≤Mi3≤Mi−13≤...≤M13≤M03=γ. |
It follows that the sequences
m1=limi→+∞mi1,M1=limi→+∞Mi1, |
m2=limi→+∞mi2,M2=limi→+∞Mi2. |
m3=limi→+∞mi3,M3=limi→+∞Mi3. |
Then
a≤m1≤M1≤b,α≤m2≤M2≤β,λ≤m3≤M3≤γ. |
By taking limits in the following equalities
mi+11=f(mi2,Mi2),Mi+11=f(Mi2,mi2), |
mi+12=g(mi3,Mi3),Mi+12=g(Mi3,mi3), |
mi+13=h(mi1,Mi1),Mi+13=h(Mi1,mi1), |
and using the continuity of
m1=f(m2,M2),M1=f(M2,m2),m2=g(m3,M3),M2=g(M3,m3),m3=h(m1,M1),M3=h(M1,m1) |
so it follows from
m1=M1,m2=M2,m3=M3. |
From
m01=a≤xn≤b=M01,m02=α≤yn≤β=M02,m03=λ≤zn≤γ=M03. |
For
m11=f(m02,M02)≤xn+1=f(yn,yn−1)≤f(M02,m02)=M11, |
m12=g(m03,M03)≤yn+1=g(zn,zn−1)≤g(M03,m03)=M12, |
and
m13=h(m01,M01)≤zn+1=h(xn,xn−1)≤h(M01,m01)=M13, |
that is
m11≤xn≤M11,m12≤yn≤M12,m13≤zn≤M13,n=3,4,⋯. |
Now, for
m21=f(m12,M12)≤xn+1=f(yn,yn−1)≤f(M12,m12)=M21, |
and
m22=g(m13,M13)≤yn+1=g(zn,zn−1)≤g(M13,m13)=M22, |
m23=h(m11,M11)≤zn+1=h(xn,xn−1)≤h(M11,m11)=M23, |
that is
m21≤xn≤M21,m22≤yn≤M22,m23≤zn≤M23,n=5,6,⋯. |
Similarly, for
m31=f(m22,M22)≤xn+1=f(yn,yn−1)≤f(M22,m22)=M31, |
m32=g(m23,M23)≤yn+1=g(zn,zn−1)≤g(M23,m23)=M32, |
and
m33=h(m21,M21)≤zn+1=h(xn,xn−1)≤h(M21,m21)=M33, |
that is
m31≤xn≤M31,m32≤yn≤M32,m33≤zn≤M33,n=7,8,⋯. |
It follows by induction that for
mi1≤xn≤Mi1,mi2≤yn≤Mi2,mi3≤zn≤Mi3,n≥2i+1. |
Using the fact that
limn→+∞xn=M1,limn→+∞yn=M2,limn→+∞zn=M3. |
From (1) and using the fact that
M1=f(M2,M2)=f(1,1),M2=g(M3,M3)=g(1,1),M3=h(M1,M1)=h(1,1). |
Theorem 2.3. Consider System (1). Assume that the following statements are true:
1.
a≤f(u,v)≤b,α≤g(u,v)≤β,λ≤h(u,v)≤γ,∀(u,v)∈(0,+∞)2. |
2.
3.
m1=f(m2,M2),M1=f(M2,m2),m2=g(m3,M3),M2=g(M3,m3),m3=h(M1,m1),M3=h(m1,M1) |
then
m1=M1,m2=M2,m3=M3. |
Then every solution of System (1) converges to the unique equilibrium point
(¯x,¯y,¯z)=(f(1,1),g(1,1),h(1,1)). |
Proof. Let
m01:=a,M01:=b,m02:=α,M02:=β,m03:=λ,M03:=γ |
and for each
mi+11:=f(mi2,Mi2),Mi+11:=f(Mi2,mi2), |
mi+12:=g(mi3,Mi3),Mi+12:=g(Mi3,mi3), |
mi+13:=h(Mi1,mi1),Mi+13:=h(mi1,Mi1). |
We have
a≤f(α,β)≤f(β,α)≤b, |
α≤g(λ,γ)≤g(γ,λ)≤β, |
λ≤h(b,a)≤h(a,b)≤γ, |
and so,
m01=a≤f(m02,M02)≤f(M02,m02)≤b=M01, |
m02=α≤g(m03,M03)≤g(M03,m03)≤β=M02, |
and
m03=λ≤h(M01,m01)≤h(m01,M01)≤γ=M03. |
Hence,
m01≤m11≤M11≤M01, |
m02≤m12≤M12≤M02, |
and
m03≤m13≤M13≤M03. |
Now, we have
m11=f(m02,M02)≤f(m12,M12)=m21≤f(M12,m12)=M21≤f(M02,m02)=M11, |
m12=g(m03,M03)≤g(m13,M13)=m22≤g(M13,m13)=M22≤g(M03,m03)=M12, |
m13=h(M01,m01)≤h(M11,m11)=m23≤h(m11,M11)=M23≤h(m01,M01)=M13, |
and it follows that
m01≤m11≤m21≤M21≤M11≤M01, |
m02≤m12≤m22≤M22≤M12≤M02, |
and
m03≤m13≤m23≤M23≤M13≤M03. |
By induction, we get for
a=m01≤m11≤...≤mi−11≤mi1≤Mi1≤Mi−11≤...≤M11≤M01=b, |
α=m02≤m12≤...≤mi−12≤mi2≤Mi2≤Mi−12≤...≤M12≤M02=β, |
and
λ=m03≤m13≤...≤mi−13≤mi3≤Mi3≤Mi−13≤...≤M13≤M03=γ. |
It follows that the sequences
m1=limi→+∞mi1,M1=limi→+∞Mi1, |
m2=limi→+∞mi2,M2=limi→+∞Mi2. |
m3=limi→+∞mi3,M3=limi→+∞Mi3. |
Then
a≤m1≤M1≤b,α≤m2≤M2≤β,λ≤m3≤M3≤γ. |
By taking limits in the following equalities
mi+11=f(mi2,Mi2),Mi+11=f(Mi2,mi2), |
mi+12=g(mi3,Mi3),Mi+12=g(Mi3,mi3), |
mi+13=h(Mi1,mi1),Mi+13=h(mi1,Mi1), |
and using the continuity of
m1=f(m2,M2),m2=f(M2,m2),m2=g(m3,M3),M2=g(M3,m3),m3=h(M1,m1),M3=h(m1,M1) |
so it follows from
m1=M1,m2=M2,m3=M3. |
From
m01=a≤xn≤b=M01,m02=α≤yn≤β=M02,m03=λ≤zn≤γ=M03. |
For
m11=f(m02,M02)≤xn+1=f(yn,yn−1)≤f(M02,m02)=M11, |
m12=g(m03,M03)≤yn+1=g(zn,zn−1)≤g(M03,m03)=M12, |
and
m13=h(M01,m01)≤zn+1=h(xn,xn−1)≤h(m01,M01)=M13, |
that is
m11≤xn≤M11,m12≤yn≤M12,m13≤zn≤M13,n=3,4,⋯. |
Now, for
m21=f(m12,M12)≤xn+1=f(yn,yn−1)≤f(M12,m12)=M21, |
and
m22=g(m13,M13)≤yn+1=g(zn,zn−1)≤g(M13,m13)=M22, |
m23=h(M11,m11)≤zn+1=h(xn,xn−1)≤h(m11,M11)=M23, |
that is
m21≤xn≤M21,m22≤yn≤M22,m23≤zn≤M23,n=5,6,⋯. |
Similarly, for
m31=f(m22,M22)≤xn+1=f(yn,yn−1)≤f(M22,m22)=M31, |
m32=g(m23,M23)≤yn+1=g(zn,zn−1)≤g(M23,m23)=M32, |
and
m33=h(M21,m21)≤zn+1=h(xn,xn−1)≤h(m21,M21)=M33, |
that is
m31≤xn≤M31,m32≤yn≤M32,m33≤zn≤M33,n=7,8,⋯. |
It follows by induction that for
mi1≤xn≤Mi1,mi2≤yn≤Mi2,mi3≤zn≤Mi3,n≥2i+1. |
Using the fact that
limn→+∞xn=M1,limn→+∞yn=M2,limn→+∞zn=M3. |
From (1) and using the fact that
M1=f(M2,M2)=f(1,1),M2=g(M3,M3)=g(1,1),M3=h(M1,M1)=h(1,1). |
Theorem 2.4. Consider System (1). Assume that the following statements are true:
1.
a≤f(u,v)≤b,α≤g(u,v)≤β,λ≤h(u,v)≤γ,∀(u,v)∈(0,+∞)2. |
2.
3.
m1=f(m2,M2),M1=f(M2,m2),m2=g(M3,m3),M2=g(m3,M3),m3=h(M1,m1),M3=h(m1,M1) |
then
m1=M1,m2=M2,m3=M3. |
Then every solution of System (1) converges to the unique equilibrium point
(¯x,¯y,¯z)=(f(1,1),g(1,1),h(1,1)). |
Proof. Let
m01:=a,M01:=b,m02:=α,M02:=β,m03:=λ,M03:=γ |
and for each
mi+11:=f(mi2,Mi2),Mi+11:=f(Mi2,mi2), |
mi+12:=g(Mi3,mi3),Mi+12:=g(mi3,Mi3), |
mi+13:=h(Mi1,mi1),Mi+13:=h(mi1,Mi1). |
We have
a≤f(α,β)≤f(β,α)≤b, |
α≤g(γ,λ)≤g(λ,γ)≤β, |
λ≤h(b,a)≤h(a,b)≤γ, |
and so,
m01=a≤f(m02,M02)≤f(M02,m02)≤b=M01, |
m02=α≤g(M03,m03)≤g(m03,M03)≤β=M02, |
and
m03=λ≤h(M01,m01)≤h(m01,M01)≤γ=M03. |
Hence,
m01≤m11≤M11≤M01, |
m02≤m12≤M12≤M02, |
and
m03≤m13≤M13≤M03. |
Now, we have
m11=f(m02,M02)≤f(m12,M12)=m21≤f(M12,m12)=M21≤f(M02,m02)=M11, |
m12=g(M03,m03)≤g(M13,m13)=m22≤g(m13,M13)=M22≤g(m03,M03)=M12, |
m13=h(M01,m01)≤h(M11,m11)=m23≤h(m11,M11)=M23≤h(m01,M01)=M13, |
and it follows that
m01≤m11≤m21≤M21≤M11≤M01, |
m02≤m12≤m22≤M22≤M12≤M02, |
and
m03≤m13≤m23≤M23≤M13≤M03. |
By induction, we get for
a=m01≤m11≤...≤mi−11≤mi1≤Mi1≤Mi−11≤...≤M11≤M01=b, |
α=m02≤m12≤...≤mi−12≤mi2≤Mi2≤Mi−12≤...≤M12≤M02=β, |
and
λ=m03≤m13≤...≤mi−13≤mi3≤Mi3≤Mi−13≤...≤M13≤M03=γ. |
It follows that the sequences
m1=limi→+∞mi1,M1=limi→+∞Mi1, |
m2=limi→+∞mi2,M2=limi→+∞Mi2. |
m3=limi→+∞mi3,M3=limi→+∞Mi3. |
Then
a≤m1≤M1≤b,α≤m2≤M2≤β,λ≤m3≤M3≤γ. |
By taking limits in the following equalities
mi+11=f(mi2,Mi2),Mi+11=f(Mi2,mi2), |
mi+12=g(Mi3,mi3),Mi+12=g(mi3,Mi3), |
mi+13=h(Mi1,mi1),Mi+13=h(mi1,Mi1), |
and using the continuity of
m1=f(m2,M2),M1=f(M2,m2),m2=g(M3,m3),M2=g(m3,M3),m3=h(M1,m1),M3=h(m1,M1) |
so it follows from
m1=M1,m2=M2,m3=M3. |
From
m01=a≤xn≤b=M01,m02=α≤yn≤β=M02,m03=λ≤zn≤γ=M03. |
For
m11=f(m02,M02)≤xn+1=f(yn,yn−1)≤f(M02,m02)=M11, |
m12=g(M03,m03)≤yn+1=g(zn,zn−1)≤g(m03,M03)=M13, |
and
m13=h(M01,m01)≤zn+1=h(xn,xn−1)≤h(m01,M01)=M13, |
that is
m11≤xn≤M11,m12≤yn≤M12,m13≤zn≤M13,n=3,4,⋯. |
Now, for
m21=f(m12,M12)≤xn+1=f(yn,yn−1)≤f(M12,m12)=M21, |
and
m22=g(M13,m13)≤yn+1=g(zn,zn−1)≤g(m13,M13)=M22, |
m23=h(M11,m11)≤zn+1=h(xn,xn−1)≤h(m11,M11)=M23, |
that is
m21≤xn≤M21,m22≤yn≤M22,m23≤zn≤M23,n=5,6,⋯. |
Similarly, for
m31=f(m22,M22)≤xn+1=f(yn,yn−1)≤f(M22,m22)=M31, |
m32=g(M23,m23)≤yn+1=g(zn,zn−1)≤g(m23,M23)=M32, |
and
m33=h(M21,m21)≤zn+1=h(xn,xn−1)≤h(m21,M21)=M33, |
that is
m31≤xn≤M31,m32≤yn≤M32,m33≤zn≤M33,n=7,8,⋯. |
It follows by induction that for
mi1≤xn≤Mi1,mi2≤yn≤Mi2,mi3≤zn≤Mi3,n≥2i+1. |
Using the fact that
limn→+∞xn=M1,limn→+∞yn=M2,limn→+∞zn=M3. |
From (1) and using the fact that
M1=f(M2,M2)=f(1,1),M2=g(M3,M3)=g(1,1),M3=h(M1,M1)=h(1,1). |
Theorem 2.5. Consider System (1). Assume that the following statements are true:
1.
a≤f(u,v)≤b,α≤g(u,v)≤β,λ≤h(u,v)≤γ,∀(u,v)∈(0,+∞)2. |
2.
3.
m1=f(m2,M2),M1=f(M2,m2),m2=g(M3,m3),M2=g(m3,M3),m3=h(m1,M1),M3=h(M1,m1) |
then
m1=M1,m2=M2,m3=M3. |
Then every solution of System (1) converges to the unique equilibrium point
(¯x,¯y,¯z)=(f(1,1),g(1,1),h(1,1)). |
Proof. Let
m01:=a,M01:=b,m02:=α,M02:=β,m03:=λ,M03:=γ |
and for each
mi+11:=f(mi2,Mi2),Mi+11:=f(Mi2,mi2), |
mi+12:=g(Mi3,mi3),Mi+12:=g(mi3,Mi3), |
mi+13:=h(mi1,Mi1),Mi+13:=h(Mi1,mi1). |
We have
a≤f(α,β)≤f(β,α)≤b, |
α≤g(γ,λ)≤g(λ,γ)≤β, |
λ≤h(a,b)≤h(b,a)≤γ, |
and so,
m01=a≤f(m02,M02)≤f(M02,m02)≤b=M01, |
m02=α≤g(M03,m03)≤g(m03,M03)≤β=M02, |
and
m03=λ≤h(m01,M01)≤h(M01,m01)≤γ=M03. |
Hence,
m01≤m11≤M11≤M01, |
m02≤m12≤M12≤M02, |
and
m03≤m13≤M13≤M03. |
Now, we have
m11=f(m02,M02)≤f(m12,M12)=m21≤f(M12,m12)=M21≤f(M02,m02)=M11, |
m12=g(M03,m03)≤g(M13,m13)=m22≤g(m13,M13)=M22≤g(m03,M03)=M12, |
m13=h(m01,M01)≤h(m11,M11)=m23≤h(M11,m11)=M23≤h(M01,m01)=M13, |
and it follows that
m01≤m11≤m21≤M21≤M11≤M01, |
m02≤m12≤m22≤M22≤M12≤M02, |
and
m03≤m13≤m23≤M23≤M13≤M03. |
By induction, we get for
a=m01≤m11≤...≤mi−11≤mi1≤Mi1≤Mi−11≤...≤M11≤M01=b, |
α=m02≤m12≤...≤mi−12≤mi2≤Mi2≤Mi−12≤...≤M12≤M02=β, |
and
λ=m03≤m13≤...≤mi−13≤mi3≤Mi3≤Mi−13≤...≤M13≤M03=γ. |
It follows that the sequences
m1=limi→+∞mi1,M1=limi→+∞Mi1, |
m2=limi→+∞mi2,M2=limi→+∞Mi2. |
m3=limi→+∞mi3,M3=limi→+∞Mi3. |
Then
a≤m1≤M1≤b,α≤m2≤M2≤β,λ≤m3≤M3≤γ. |
By taking limits in the following equalities
mi+11=f(mi2,Mi2),Mi+11=f(Mi2,mi2), |
mi+12=g(Mi3,mi3),Mi+12=g(mi3,Mi3), |
mi+13=h(mi1,Mi1),Mi+13=h(Mi1,mi1), |
and using the continuity of
m1=f(m2,M2),M1=f(M2,m2),m2=g(M3,m3),M2=g(m3,M3),m3=h(m1,M1),M3=h(M1,m1) |
so it follows from
m1=M1,m2=M2,m3=M3. |
From
m01=a≤xn≤b=M01,m02=α≤yn≤β=M02,m03=λ≤zn≤γ=M03. |
For
m11=f(m02,M02)≤xn+1=f(yn,yn−1)≤f(M02,m02)=M11, |
m12=g(M03,m03)≤yn+1=g(zn,zn−1)≤g(m03,M03)=M13, |
and
m13=h(m01,M01)≤zn+1=h(xn,xn−1)≤h(M01,m01)=M13, |
that is
m11≤xn≤M11,m12≤yn≤M12,m13≤zn≤M13,n=3,4,⋯. |
Now, for
m21=f(m12,M12)≤xn+1=f(yn,yn−1)≤f(M12,m12)=M21, |
and
m22=g(M13,m13)≤yn+1=g(zn,zn−1)≤g(m13,M13)=M22, |
m23=h(m11,M11)≤zn+1=h(xn,xn−1)≤h(M11,m11)=M23, |
that is
m21≤xn≤M21,m22≤yn≤M22,m23≤zn≤M23,n=5,6,⋯. |
Similarly, for
m31=f(m22,M22)≤xn+1=f(yn,yn−1)≤f(M22,m22)=M31, |
m32=g(M23,m23)≤yn+1=g(zn,zn−1)≤g(m23,M23)=M32, |
and
m33=h(m21,M21)≤zn+1=h(xn,xn−1)≤h(M21,m21)=M33, |
that is
m31≤xn≤M31,m32≤yn≤M32,m33≤zn≤M33,n=7,8,⋯. |
It follows by induction that for
mi1≤xn≤Mi1,mi2≤yn≤Mi2,mi3≤zn≤Mi3,n≥2i+1. |
Using the fact that
limn→+∞xn=M1,limn→+∞yn=M2,limn→+∞zn=M3. |
From (1) and using the fact that
M1=f(M2,M2)=f(1,1),M2=g(M3,M3)=g(1,1),M3=h(M1,M1)=h(1,1). |
Theorem 2.6. Consider System (1). Assume that the following statements are true:
1.
a≤f(u,v)≤b,α≤g(u,v)≤β,λ≤h(u,v)≤γ,∀(u,v)∈(0,+∞)2. |
2.
3.
m1=f(M2,m2),M1=f(m2,M2),m2=g(M3,m3),M2=g(m3,M3),m3=h(M1,m1),M3=h(m1,M1) |
then
m1=M1,m2=M2,m3=M3. |
Then every solution of System (1) converges to the unique equilibrium point
(¯x,¯y,¯z)=(f(1,1),g(1,1),h(1,1)). |
Proof. Let
m01:=a,M01:=b,m02:=α,M02:=β,m03:=λ,M03:=γ |
and for each
mi+11:=f(Mi2,mi2),Mi+11:=f(mi2,Mi2), |
mi+12:=g(Mi3,mi3),Mi+12:=g(mi3,Mi3), |
mi+13:=h(Mi1,mi1),Mi+13:=h(mi1,Mi1). |
We have
a≤f(β,α)≤f(α,β)≤b, |
α≤g(γ,λ)≤g(λ,γ)≤β, |
λ≤h(b,a)≤h(a,b)≤γ |
and so,
m01=a≤f(M02,m02)≤f(m02,M02)≤b=M01, |
m02=α≤g(M03,m03)≤g(m03,M03)≤β=M02, |
m03=λ≤h(M01,m01)≤h(m01,M01)≤γ=M03. |
Hence,
m01≤m11≤M11≤M01, |
m02≤m12≤M12≤M02, |
and
m03≤m13≤M13≤M03. |
Now, we have
m11=f(M02,m02)≤f(M12,m12)=m21≤f(m12,M12)=M21≤f(m02,M02)=M11, |
m12=g(M03,m03)≤g(M13,m13)=m22≤g(m13,M13)=M22≤g(m03,M03)=M12, |
m13=h(M01,m01)≤h(M11,m11)=m23≤h(m11,M11)=M23≤h(m01,M01)=M13 |
and it follows that
m01≤m11≤m21≤M21≤M11≤M01, |
m02≤m12≤m22≤M22≤M12≤M02, |
m03≤m13≤m23≤M23≤M13≤M03. |
By induction, we get for
a=m01≤m11≤...≤mi−11≤mi1≤Mi1≤Mi−11≤...≤M11≤M01=b, |
α=m02≤m12≤...≤mi−12≤mi2≤Mi2≤Mi−12≤...≤M12≤M02=β, |
and
λ=m03≤m13≤...≤mi−13≤mi3≤Mi3≤Mi−13≤...≤M13≤M03=γ. |
It follows that the sequences
m1=limi→+∞mi1,M1=limi→+∞Mi1, |
m2=limi→+∞mi2,M2=limi→+∞Mi2. |
m3=limi→+∞mi3,M3=limi→+∞Mi3. |
Then
a≤m1≤M1≤b,α≤m2≤M2≤β,λ≤m3≤M3≤γ. |
By taking limits in the following equalities
mi+11=f(Mi2,mi2),Mi+11=f(mi2,Mi2), |
mi+12=g(Mi3,mi3),Mi+12=g(mi3,Mi3), |
mi+13=h(Mi1,mi1),Mi+13=h(mi1,Mi1) |
and using the continuity of
m1=f(M2,m2),M1=f(m2,M2),m2=g(M3,m3),M2=g(m3,M3),m3=h(M1,m1),M3=h(m1,M1) |
so it follows from
m1=M1,m2=M2,m3=M3. |
From
m01=a≤xn≤b=M01,m02=α≤yn≤β=M02,m03=λ≤zn≤γ=M03. |
For
m11=f(M02,m02)≤xn+1=f(yn,yn−1)≤f(m02,M02)=M11, |
m12=g(M03,m03)≤yn+1=g(zn,zn−1)≤g(m03,M03)=M12, |
m13=h(M01,m01)≤zn+1=h(xn,xn−1)≤h(m01,M01)=M13, |
that is
m11≤xn≤M11,m12≤yn≤M12,m13≤zn≤M13,n=3,4,⋯. |
Now, for
m21=f(M12,m12)≤xn+1=f(yn,yn−1)≤f(m12,M12)=M21, |
m22=g(M13,m31)≤yn+1=g(zn,zn−1)≤g(m13,M31)=M22, |
m23=h(M11,m11)≤zn+1=h(xn,xn−1)≤h(m11,M11)=M23, |
that is
m21≤xn≤M21,m22≤yn≤M22,m23≤zn≤M23,n=5,6,⋯. |
Similarly, for
m31=f(M22,m22)≤xn+1=f(yn,yn−1)≤f(m22,M22)=M31, |
m32=g(M23,m23)≤yn+1=g(zn,zn−1)≤g(m23,M23)=M32, |
m33=h(M21,m21)≤zn+1=h(xn,xn−1)≤h(m21,M21)=M33, |
that is
m31≤xn≤M31,m32≤yn≤M32,m33≤zn≤M33,n=7,8,⋯. |
It follows by induction that for
mi1≤xn≤Mi1,mi2≤yn≤Mi2,mi3≤zn≤Mi3,n≥2i+1. |
Using the fact that
limn→+∞xn=M1,limn→+∞yn=M2,limn→+∞zn=M3. |
From (1) and using the fact that
M1=f(M2,M2)=f(1,1),M2=g(M3,M3)=g(1,1),M3=h(M1,M1)=h(1,1). |
Theorem 2.7. Consider System (1). Assume that the following statements are true:
1.
a≤f(u,v)≤b,α≤g(u,v)≤β,λ≤h(u,v)≤γ,∀(u,v)∈(0,+∞)2. |
2.
3.
m1=f(M2,m2),M1=f(m2,M2),m2=g(M3,m3),M2=g(m3,M3),m3=h(m1,M1),M3=h(M1,m1) |
then
m1=M1,m2=M2,m3=M3. |
Then every solution of System (1) converges to the unique equilibrium point
(¯x,¯y,¯z)=(f(1,1),g(1,1),h(1,1)). |
Proof. Let
m01:=a,M01:=b,m02:=α,M02:=β,m03:=λ,M03:=γ |
and for each
mi+11:=f(Mi2,mi2),Mi+11:=f(mi2,Mi2), |
mi+12:=g(Mi3,mi3),Mi+12:=g(mi3,Mi3), |
mi+13:=h(mi1,Mi1),Mi+13:=h(Mi1,mi1). |
We have
a≤f(β,α)≤f(α,β)≤b, |
α≤g(γ,λ)≤g(λ,γ)≤β, |
λ≤h(a,b)≤h(b,a)≤γ, |
and so,
m01=a≤f(M02,m02)≤f(m02,M02)≤b=M01, |
m02=α≤g(M03,m03)≤g(m03,M03)≤β=M02, |
and
m03=λ≤h(m01,M01)≤h(M01,m01)≤γ=M03. |
Hence,
m01≤m11≤M11≤M01, |
m02≤m12≤M12≤M02, |
and
m03≤m13≤M13≤M03. |
Now, we have
m11=f(M02,m02)≤f(M12,m12)=m21≤f(m12,M12)=M21≤f(m02,M02)=M11, |
m12=g(M03,m03)≤g(M13,m13)=m22≤g(m13,M13)=M22≤g(m03,M03)=M12, |
m13=h(m01,M01)≤h(m11,M11)=m23≤h(M11,m11)=M23≤h(M01,m01)=M13, |
and it follows that
m01≤m11≤m21≤M21≤M11≤M01, |
m02≤m12≤m22≤M22≤M12≤M02, |
and
m03≤m13≤m23≤M23≤M13≤M03. |
By induction, we get for
a=m01≤m11≤...≤mi−11≤mi1≤Mi1≤Mi−11≤...≤M11≤M01=b, |
α=m02≤m12≤...≤mi−12≤mi2≤Mi2≤Mi−12≤...≤M12≤M02=β, |
and
λ=m03≤m13≤...≤mi−13≤mi3≤Mi3≤Mi−13≤...≤M13≤M03=γ. |
It follows that the sequences
m1=limi→+∞mi1,M1=limi→+∞Mi1, |
m2=limi→+∞mi2,M2=limi→+∞Mi2. |
m3=limi→+∞mi3,M3=limi→+∞Mi3. |
Then
a≤m1≤M1≤b,α≤m2≤M2≤β,λ≤m3≤M3≤γ. |
By taking limits in the following equalities
mi+11=f(Mi2,mi2),Mi+11=f(mi2,Mi2), |
mi+12=g(Mi3,mi3),Mi+12=g(mi3,Mi3), |
mi+13=h(mi1,Mi1),Mi+13=h(Mi1,mi1), |
and using the continuity of
m1=f(M2,m2),M1=f(m2,M2),m2=g(M3,m3),M2=g(m3,M3),m3=h(m1,M1),M3=h(M1,m1) |
so it follows from
m1=M1,m2=M2,m3=M3. |
From
m01=a≤xn≤b=M01,m02=α≤yn≤β=M02,m03=λ≤zn≤γ=M03. |
For
m11=f(M02,m02)≤xn+1=f(yn,yn−1)≤f(m02,M02)=M11, |
m12=g(M03,m03)≤yn+1=g(zn,zn−1)≤g(m03,M03)=M12, |
and
m13=h(m01,M01)≤zn+1=h(xn,xn−1)≤h(M01,m01)=M13, |
that is
m11≤xn≤M11,m12≤yn≤M12,m13≤zn≤M13,n=3,4,⋯. |
Now, for
m21=f(M12,m12)≤xn+1=f(yn,yn−1)≤f(m12,M12)=M21, |
m22=g(M13,m31)≤yn+1=g(zn,zn−1)≤g(m13,M31)=M22, |
and
m23=h(m11,M11)≤zn+1=h(xn,xn−1)≤h(M11,m11)=M23, |
that is
m21≤xn≤M21,m22≤yn≤M22,m23≤zn≤M23,n=5,6,⋯. |
Similarly, for
m31=f(M22,m22)≤xn+1=f(yn,yn−1)≤f(m22,M22)=M31, |
m32=g(M23,m23)≤yn+1=g(zn,zn−1)≤g(m23,M23)=M32, |
and
m33=h(m21,M21)≤zn+1=h(xn,xn−1)≤h(M21,m21)=M33, |
that is
m31≤xn≤M31,m32≤yn≤M32,m33≤zn≤M33,n=7,8,⋯. |
It follows by induction that for
mi1≤xn≤Mi1,mi2≤yn≤Mi2,mi3≤zn≤Mi3,n≥2i+1. |
Using the fact that
limn→+∞xn=M1,limn→+∞yn=M2,limn→+∞zn=M3. |
From (1) and using the fact that
M1=f(M2,M2)=f(1,1),M2=g(M3,M3)=g(1,1),M3=h(M1,M1)=h(1,1). |
Theorem 2.8. Consider System (1). Assume that the following statements are true:
1.
a≤f(u,v)≤b,α≤g(u,v)≤β,λ≤h(u,v)≤γ,∀(u,v)∈(0,+∞)2. |
2.
3.
m1=f(M2,m2),M1=f(m2,M2),m2=g(m3,M3),M2=g(M3,m3),m3=h(m1,M1),M3=h(M1,m1) |
then
m1=M1,m2=M2,m3=M3. |
Then every solution of System (1) converges to the unique equilibrium point
(¯x,¯y,¯z)=(f(1,1),g(1,1),h(1,1)). |
Proof. Let
m01:=a,M01:=b,m02:=α,M02:=β,m03:=λ,M03:=γ |
and for each
mi+11:=f(Mi2,mi2),Mi+11:=f(mi2,Mi2), |
mi+12:=g(mi3,Mi3),Mi+12:=g(Mi3,mi3), |
mi+13:=h(mi1,Mi1),Mi+13:=h(Mi1,mi1). |
We have
a≤f(β,α)≤f(α,β)≤b, |
α≤g(λ,γ)≤g(γ,λ)≤β, |
λ≤h(a,b)≤h(b,a)≤γ, |
and so,
m01=a≤f(M02,m02)≤f(m02,M02)≤b=M01, |
m02=α≤g(m03,M03)≤g(M03,m03)≤β=M02, |
and
m03=λ≤h(m01,M01)≤g(M01,m01)≤γ=M03. |
Hence,
m01≤m11≤M11≤M01, |
m02≤m12≤M12≤M02, |
and
m03≤m13≤M13≤M03. |
Now, we have
m11=f(M02,m02)≤f(M12,m12)=m21≤f(m12,M12)=M21≤f(m02,M02)=M11, |
and it follows that
and
By induction, we get for
and
It follows that the sequences
Then
By taking limits in the following equalities
and using the continuity of
so it follows from
From
For
and
that is
Now, for
and
that is
Similarly, for
and
that is
It follows by induction that for
Using the fact that
From (1) and using the fact that
Theorem 2.9. Consider System . Assume that the following statements are true:
1.
2.
3.
then
Then every solution of System converges to the unique equilibrium point
Proof. Let
and for each
We have
and so,
and
Hence,
and
Now, we have
and it follows that
and
By induction, we get for
and
It follows that the sequences
Then
By taking limits in the following equalities
and using the continuity of
so it follows from
From
For
and
that is
Now, for
and
that is
Similarly, for
and
that is
It follows by induction that for
Using the fact that
From (1) and using the fact that
The following theorem is devoted to global stability of the equilibrium point
Theorem 2.10. Under the hypotheses of Theorem 2.1 and one of Theorems 2.2– 2.9, the equilibrium point
Now as an application of the previous results, we give an example.
Example 1. Consider the following system of difference equations
(6) |
where
Assume that
It follows from Theorem 2.1 that the unique equilibrium point
of System (6) will be locally stable if
which is equivalent to
Also, we have conditions
For this purpose, let
(7) |
(8) |
(9) |
From (7)-(9), we get
(10) |
(11) |
(12) |
Now, from (10)-(12), we get
(13) |
where
It is not hard to see that
Using the fact
we get
If we choose the parameters
then we get that
and so it follows from (13) that
Using this and (10)-(12), we obtain
In summary we have the following result.
Theorem 2.11. Assume that that parameters
●
●
●
Then the equilibrium point
We visualize the solutions of System (6) in Figures 1-3. In Figure 1 and Figure 2, we give the solution and corresponding global attractor of System (6) for
However Figure 3 shows the unstable solution corresponding to the values
Here, we are interested in existence of periodic solutions for System (1). In the following result we will established a necessary and sufficient condition for which there exist prime period two solutions of System (1).
Theorem 3.1. Assume that
if and only if
Proof. Let
is a solution for System (1). Then, we have
(14) |
(15) |
(16) |
(17) |
(18) |
(19) |
From (14)-(19), it follows that
Now, assume that
and let
We have
By induction we get
In the following, we apply our result in finding prime period two solutions of two special Systems.
Consider the three dimensional system of difference equations
(20) |
where
System (20) can be seen as a generalization of the system
studied in [23]. This last one is also a generalization of the equation
Corollary 1. Assume that
if and only if
(21) |
Proof. System (20) can be written as
where
So, from Theorem 3.1,
will be a period prime two solution of System (20) if and only if
Clearly this condition is equivalent to (21).
Example 2. If we choose
The last condition is satisfied for the choice
of the parameters. The corresponding prime period two solution will be
and
that is
Now, consider the following system of difference equations
(22) |
where the initial values
Corollary 2. Assume that
if and only if
(23) |
Proof. System (22) can be written as
where
So, from Theorem 3.1,
will be a period prime two solution of System (20) if and only if
Clearly this condition is equivalent to
Example 3. For
The last condition is satisfied for the choice
of the parameters. The corresponding prime period two solution will be
and
that is
Here, we are interested in the oscillation of the solutions of System (1) about the equilibrium point
Theorem 4.1. Let
1. If
then we get
That is for the sequences
2. If
then we get
That is for the sequences
Proof. 1. Assume that
We have
By induction, we get
2. Assume that
We have
By induction, we get
So, the proof is completed. Now, we will apply the results of this section on the following particular system.
Example 4. Consider the system of difference equations
(24) |
where
Let
It is not hard to see that
System (24) has the unique equilibrium point
Corollary 3. Let
1. Let
Then the sequences
2. Let
Then the sequences
Proof. 1. Let
We have
which implies that
Using the fact that
we get
Also as,
we get
Now, as
we obtain
Similarly,
and
and by induction we get that
for
and this is the statement of Part 1. of Theorem 4.1.
We have
which implies that
Using the fact that
we get
Also as,
we get
Now, as
we obtain
Similarly,
and
Thus, by induction we get that
for
and this is the statement of Part 2. of Theorem 4.1.
In this study, the global stability of the unique positive equilibrium point of a three-dimensional general system of difference equations defined by positive and homogeneous functions of degree zero was studied. For this, general convergence theorems were given considering all possible monotonicity cases in arguments of functions
The authors thanks the two referees for their comments and suggestions. The work of N. Touafek and Y. Akrour was supported by DGRSDT (MESRS-DZ).
[1] |
Block decomposition of permutations and Schur-positivity. J. Algebraic Combin. (2018) 47: 603-622. ![]() |
[2] | C. A. Athanasiadis, Gamma-positivity in combinatorics and geometry, Sém. Lothar. Combin., 77 (2018), Article B77i, 64 pp (electronic). |
[3] | M. Barnabei, F. Bonetti and M. Silimbani, The descent statistic on -avoiding permutations, Sém. Lothar. Combin., 63 (2010), B63a, 8 pp. |
[4] |
Kazhdan-Lusztig polynomials for -Hexagon-avoiding permutations. J. Algebraic Combin. (2001) 13: 111-136. ![]() |
[5] |
M. Bóna, Combinatorics of Permutations. With a Foreword by Richard Stanley, Discrete Mathematics and its Applications (Boca Raton). Chapman & Hall/CRC, Boca Raton, FL, 2004. doi: 10.1201/9780203494370
![]() |
[6] |
The number of Baxter permutations. J. Combin. Theory Ser. A (1978) 24: 382-394. ![]() |
[7] | A. Claesson and S. Kitaev, Classification of bijections between -and -avoiding permutations, Sém. Lothar. Combin., 60 (2008), B60d, 30 pp. |
[8] |
Decompositions and statistics for -trees and nonseparable permutations. Adv. in Appl. Math. (2009) 42: 313-328. ![]() |
[9] | L. Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions, Revised and enlarged edition, D. Reidel Publishing Co., Dordrecht, 1974. |
[10] | S. Corteel, M. A. Martinez, C. D. Savage and M. Weselcouch, Patterns in inversion sequences I, Discrete Math. Theor. Comput. Sci., 18 (2016), Paper No. 2, 21 pp. |
[11] |
Permutation patterns and statistics. Discrete Math. (2012) 312: 2760-2775. ![]() |
[12] | P. G. Doyle, Stackable and queueable permutations, preprint, arXiv: 1201.6580. |
[13] |
Bijections for refined restricted permutations. J. Combin. Theory Ser. A (2004) 105: 207-219. ![]() |
[14] | D. Foata and M.-P. Schützenberger, Théorie Géométrique des Polynômes Eulériens, Lecture Notes in Mathematics, Vol. 138, Springer-Verlag, Berlin-New York, 1970. |
[15] |
-arrangements, statistics, and patterns. SIAM J. Discrete Math. (2020) 34: 1830-1853. ![]() |
[16] | S. Fu, Z. Lin and Y. Wang, A combinatorial bijection on di-sk trees, preprint, arXiv: 2011.11302. |
[17] |
On two new unimodal descent polynomials. Discrete Math. (2018) 341: 2616-2626. ![]() |
[18] |
S. Fu and Y. Wang, Bijective proofs of recurrences involving two Schröder triangles, European J. Combin., 86 (2020), 103077, 18 pp. doi: 10.1016/j.ejc.2019.103077
![]() |
[19] | A. L. L. Gao, S. Kitaev and P. B. Zhang, On pattern avoiding indecomposable permutations, Integers, 18 (2018), A2, 23 pp. |
[20] |
S. Kitaev, Patterns in Permutations and Words. With a Forewrod by Jeffrey B. Remmel, Monographs in Theoretical Computer Science. An EATCS Series. Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-17333-2
![]() |
[21] | D. E. Knuth, The Art of Computer Programming. Vol. 1. Fundamental Algorithms, 3 edition, Addison-Wesley, Reading, MA, 1997. |
[22] |
Permutations with restricted patterns and Dyck paths. Special Issue in Honor of Dominique Foata's 65th birthday, Adv. in Appl. Math. (2001) 27: 510-530. ![]() |
[23] |
On -positive polynomials arising in pattern avoidance. Adv. in Appl. Math. (2017) 82: 1-22. ![]() |
[24] |
A sextuple equidistribution arising in pattern avoidance. J. Combin. Theory Ser. A (2018) 155: 267-286. ![]() |
[25] | OEIS Foundation Inc., The On-Line Encyclopedia of Integer Sequences, http://oeis.org, 2020. |
[26] |
T. K. Petersen, Eulerian numbers. With a Foreword by Richard Stanley, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser/Springer, New York, 2015. doi: 10.1007/978-1-4939-3091-3
![]() |
[27] | M. Rubey, An involution on Dyck paths that preserves the rise composition and interchanges the number of returns and the position of the first double fall, Sém. Lothar. Combin., 77 (2016-2018), Art. B77f, 4 pp. |
[28] |
Restricted permutations. European J. Combin. (1985) 6: 383-406. ![]() |
[29] |
Forbidden subsequences. Discrete Math. (1994) 132: 291-316. ![]() |
[30] |
The Eulerian distribution on involutions is indeed -positive. J. Combin. Theory Ser. A (2019) 165: 139-151. ![]() |
[31] |
Generating trees and the Catalan and Schröder numbers. Discrete Math. (1995) 146: 247-262. ![]() |