The asymptotic properties of Eulerian numbers and refined Eulerian numbers: A Spline perspective

0.16 MB·English

by  Renhong Wang

The asymptotic properties of Eulerian numbers
and refined Eulerian numbers: A Spline
perspective
∗
0
1 Renhong Wang Yan Xu
† ‡
0
2 Instituteof Mathematical Sciences, Dalian University of Technology,

n
Dalian, 116024, China
a
J
Abstract
0
3 In this paper, the asymptotic formulas for Eulerian numbers, refined
Euleriannumbersandthecoefficientsofdescentpolynomialsareobtained
]
directly from thespline interpretations of these numbers. Having related
O
these numbers directly to B-splines [15], we can take advantage of many
C
powerfulsplinetechniquestoderivevariousresultsofthesenumbers. The
h. asymptotic formulas for the Eulerian numbers Ad,k agree with the pre-
t viously known results which were given by L. Carlitz et al.(1972)[2] and
a
S.Tanny (1973) [18], but the convergence order is much better. We also
m
give the asymptotic representations of refined Eulerian numberswhich is
[ in terms of the Hermitepolynomials.

1
v Keywords: B-splines;Euleriannumbers;RefinedEuleriannumbers;Descent
6
5 polynomials; Asymptotic approximation.
0
0
.
2
0 1 Introduction
0
1
: Eulerian number, denoted here by A is the number of permutations in the
v d,k
i
X symmetricgroupS thathaveexactlyk 1descents. Theyplayimportantroles
d
−
r
a inenumerativecombinatorics,geometry[12],statisticalapplications[2,18],and
spline theory[10, 15]. The asymptotic properties of Eulerian numbers was ex-
amined by S. Tanny [18] and L. Carlitz et. al.[2] with the help of central Limit
∗ThisworkwaspartiallySupportedbyTheNationalNaturalScienceFoundationofChina
(NO.60373093, NO.60533060, NO.10801024, U0935004 and NO.10726068.) and the Innova-
tionFundationoftheKeyLaboratoryofHighTemperatureGasdynamicsofChineseAcademy
ofSciences. ThisresearchprojectwascarriedwhilethesecondauthorvisitedMichiganState
University in 2008-09 under the support of State Scholarship Fund by China Scholarship
Councilin2008.
†Email: [e-mail]
‡Email: [e-mail]
1
theoremofprobabilitytheory. Theyshowedthatthe Euleriannumbersapprox-
imate to the Gaussian function with the convergence order O(d−43). Neverthe-
less,inthis paper,we obtainamuchbetter convergenceorderO(d−32)with the
help of the spline theory.
The refined Eulerian number A is the number of permutations in the
d,k,j
symmetricgroupS withk descentsandendingwith the elementj. Brentiand
d
Welker [9] gave a precise description of the h-vector of the barycentric subdi-
vision of a simplicial complex in terms of the h-vector of the original complex
and the refined Eulerian numbers. Later, Martina Kubitzke and Eran Nevo
[13] derived new inequalities on A by the algebraic method. A result due
d,k,j
to Ehrenborg, Readdy and Steingr´ımsson[17] gave a geometry interpolation of
A as the mixed volumes of two adjacent slices from the unit cube. In [15]
d,k,j
Renhong,W., Yan,X. andZhiqiang,X. gavea spline interpolationofthe A
d,k,j
by the geometry interpolation and the spline theory. Further more, they gave
the explicit and recurrence representation of A . In [11], the experimen-
d,k,j
tal evidence suggested that the refined Eulerian number A is not exactly
d,k,j
normal approximation as d grows large. In this paper, we give the asymptotic
representations of refined Eulerian numbers in terms of the Hermite polynomi-
als.
Recently, Steingr´ımsson [7] generalized the definition of descents and ex-
cedancestotheelements(calledindexedpermutations,coloredpermutationsor
r-signedpermutationssee[17,7,5,4])ofgroupsSn :=Z S ,where iswreath
d n≀ d ≀
productwithrespecttotheusualactionofS bypermutationsof[d]. Thenum-
d
ber of indexed permutations in Sn with k descents is denoted by D(d,n,k). In
d
[7], using the work of Brenti [8], Steingr´ımsson [7] showed that the numbers
D(d,n,k) are unimodal. Moreover, In[15], using the spline theory, the authors
gavetheexplicitexpressionofD(d,n,k)andshowedthe numbersD(d,n,k)are
log-concave. In this paper, we show that D(d,n,k) is approximately normal in
terms of the spline interpretation of D(d,n,k).
2
The paper is organized as follows. In Section 2, after recalling some neces-
sary definitions and notations, we show the connection between the B-splines
and these combinatorial numbers. In Section 3, we give the main results of
this paper. The results show the development of various asymptotic properties
of Eulerian numbers, refined Eulerian numbers and the coefficients of descent
polynomials,whichareobtaineddirectlyfromthesplineinterpretationsofthese
numbers. We prove the main results in Section 4.
2 definitions and notations
Eulerian number, denoted here by A is the number of permutations in the
d,k
symmetric group S that have exactly k 1 descents. An recurrence formula
d
−
for A is
d,k
A =(k+1)A +(d k)A (2.1)
d,k+1 d 1,k+1 d 1,k
− − −
with the boundary conditions
A =1,A =0,d>0.
0,0 d,0
An explicit formula for A is
d,k
k
d+1
A = ( 1)i(k i)d (2.2)
d,k
i − −
i=0(cid:18) (cid:19)
X
which can be easily verified using the above recurrence.
Refined Eulerian number A is the number of permutations in the sym-
d,k,j
metricgroupS withk descentsandendingwiththeelementj. Therecurrence
d
formulas for A are [15]
d,k,j
A =(k+1)A +(d k)A ,
d+1,k,d j+1 d,k,d j d,k 1,d j
− − − − −
(2.3)
A =kA +(d k+1)A .
d+1,k,d j+1 d,k,d j+1 d,k 1,d j+1
− − − − −
An explicit formula for A is [15]
d,k,j
k
d
Ad,k,j = ( 1)i(k i)d−j(k i+1)j−1.
i − − −
i=0(cid:18) (cid:19)
X
3
Descent polynomials, denoted by Dn(t), are defined as
d
d
Dn(t)= D(d,n,k)tk,
d
k=0
X
where D(d,n,k) is the number of indexed permutations in Sn with k descents.
d
The indexed permutation of length d and with indices in 0,1,...,n 1 is an
{ − }
ordinary permutation in the symmetric group S where each letter has been
d
assigned an integer between 0 and n 1. Indexed permutations, or r-signed
−
permutations, are a generalizationof permutations (see [17, 7]). We will follow
the notation in [7]. The set of all such indexed permutations is denoted by Sn.
d
The numbers D(d,n,k) satisfy a simple three-term recurrence[7],
D(d,n,k)=(nk+1)D(d 1,n,k)+(n(d k)+(n 1))D(d 1,n,k 1). (2.4)
− − − − −
And also has an explicit formula [15]:
k
d+1
D(d,n,k)= ( 1)i(n(k i)+1)d.
i − −
i=0(cid:18) (cid:19)
X
We turnto the definitionsofB-splinesandHermite polynomials,thenshowthe
relations among Eulerian numbers, the refined Eulerian numbers, the descent
polynomials and B-splines.
Let p R 1 and Lp(R) as usual denote the set
≥
∈
+
Lp(R)= f :R Cf measurable, ∞ f(t)pdt< .
{ → | | | ∞}
Z−∞
For p=2 and f,g L2(R) define the inner product
∈
+
∞
f,g := f(t)g(t)dt
h i
Z−∞
and the norm
f := f,f
k k h i
p
making L2(R) to a Hilbert space.
4
For f L1(R) define the Fourier transform f and the inverse Fourier
∧
∈
transform f as
∨
+
f (ω) := ∞f(t)e iωtdt,
∧ −
Z−∞
f (ω) := 1 +∞f(t)e iωtdt.
∨ −
2π
Z−∞
This definition can be extended to functions f L2(R), see for example [6].
∈
B-splines with order d,whichisdenotedasB (),isdefinedbytheinduction
d
·
as
1 if x [0,1),
B (x) = ∈
1
(0 otherwise,
and for d 1
≥
B = B B ,
d 1 d 1
∗ −
where denotes the operation of convolution which is defined:
∗
+
∞
(f g)(t):= f(t y)g(y)dy
∗ −
Z−∞
for f and g in L2(R).
Then B has the compact support [0,d] and is in Cd 1(R). A well known
d −
explicit formula for B () is
d
·
d
1 d
Bd(x)= (d 1)! (−1)i i (x−i)d+−1 (2.5)
− i=0 (cid:18) (cid:19)
X
where the one-sided power function is defined by:
xd x 0,
xd = ≥
+ 0 otherwise.
(cid:26)
And the recurrence relation is
x d x
B (x)= B (x)+ − B (x 1). (2.6)
d d 1 d 1
d 1 − d 1 − −
− −
For all B , d 1, it holds B L1(R) L2(R) and
d d
≥ ∈
T
ω
B (ω)=sincd (2.7)
d∧ 2
(cid:16) (cid:17)
5
where sinc(t) denotes:
sint t=0,
sinc(t):= t 6 (2.8)
1 t=0.
(cid:26)
B-splines play important roles in approximation, computer-aid design(CAD),
signalprocessingand discrete geometry. They havebeen welldeveloped during
the past few decades. For extensive monographs see [3, 19].
Hermite polynomials H (x) with degree d, are set of orthogonal polynomi-
n
als over the domain ( ,+ ) with weighting function e x2. The Hermite
−
−∞ ∞
polynomials are defined by
Hn(x)=(−1)nex22 ddxnne−2x2 (2.9)
and satisfy
[n]
2 ( 1)kn!(2x)n 2k
−
H (x) = − .
n
k!(n 2k)!
k=0 −
X
Being the limiting case of several families of classical orthogonal polynomials,
they are of fundamental importance in asymptotic analysis.
In this paper, we shall be concerned primarily with the development of var-
ious asymptotic properties of Eulerian numbers, refined Eulerian numbers and
the coefficients of descent polynomials, which are obtained directly from the
spline interpretations of these numbers [15]. The theory of B-splines is a well
developed area of applied numerical analysis and interpolation theory. Having
relatedthese numbersdirectlytoB-splines[15],wecantakeadvantageofmany
powerful spline techniques to derive various results of these numbers. It is pre-
cisely this fact which motivates the need for more purely spline interpretations
of these numbers to which we now turn. To state conveniently, we use [λj]f(λ)
to denote the coefficient of λj in f(λ) for any given power series f(λ).
Lemma 2.1. [15]
(i) A = d! B (k);
d,k d+1
·
6
1
(ii) D(d,n,k) = d! nd B k+ ;
d+1
· · n
(cid:18) (cid:19)
1 d
(iii) A = d! [λj] (λ+1)dB k+ / , λ 0.
d+1,k,d j+1 d+1
− · λ+1 j ≥
(cid:18) (cid:18) (cid:19)(cid:19) (cid:18) (cid:19)
3 Main results
After the necessary definitions have been provided, we can come to the main
results of this article.
Theorem 3.1. Let be k N then for d > k + 2 the sequence of the k-th
∈
derivatives B(k) of the B-spline converges to the k-th derivative of the Gaussian
d
function,
k+1
d 2 d d 1 x2 1
B(k) x+ = Dkexp +O , (3.1)
(cid:18)12(cid:19) d r12 2! √2π (cid:18)− 2 (cid:19) (cid:18)d(cid:19)
and
k+1
d 2 d d 1 x2
lim B(k) x+ = Dkexp , (3.2)
d→∞((cid:18)12(cid:19) d r12 2!) √2π (cid:18)− 2 (cid:19)
where the limit may be taken point-wise or in Lp(R),p [2, ).
∈ ∞
Sommerfeldin1904[1]showthattheGaussianfunctioncanbeapproximated
by B splines point-wise. In 1992, Unser and colleagues[14] proved that the
sequence of normalizedand scaled B-splines B tends to the Gaussianfunction
d
as the order d increases in Lp(R). A result due to Ralph Brinks [16] generalize
Unser’s result to the derivatives of the B-splines. In this paper, we reprove the
theorem and give the order of the convergence.
Theorem 3.2. For x = d+1x+ d+1, we have
d 12 2
q
1 6 x2
3
d!Ad,[xd] =sπ(d+1)exp − 2 +O d−2 (3.3)
(cid:18) (cid:19) (cid:16) (cid:17)
L. Carlitz, D. C. Kurtz, R. Scoville and O. P. Stackelberg in [2] showed
(3.3) with the help of the central limit theorem of probability theory that the
−3
expression on the right side of (3.3) has the order O(d 4 ). Nevertheless, using
7
the spline interpretation of Eulerian numbers, we obtain the same asymptotic
forms of An,k with a much better convergence order O(d−32).
Theorem 3.3. For x = d+1x+ d+1, we have
d 12 2
q
1 6 x+ 1 2
d!·ndD(d,n,[xd])=sπ(d+1)exp −(cid:0) 2n(cid:1) !+O(cid:16)d−32(cid:17) (3.4)
Using the spline theory, we can also get the asymptotic representations of
refined Eulerian numbers in terms of the Hermite polynomials.
Theorem 3.4. Let x = d+1(x 1)+ d+1 then
d 12 − 2
q
6 x2 j 1 d+1 −2i −3
Ad+1,[xd],d−j+1 = d!sπ(d+1)exp(cid:18)− 2 (cid:19)Xi=0 d−ij+i (cid:18) 12 (cid:19) Hi(x)+O(cid:16)d 2 (cid:17),
(cid:0) (cid:1)
where H (x) are the Hermite polynomials are defined by
n
Hn(x)=(−1)nex22 ddxnne−2x2. (3.5)
4 Proofs of the main results
TheproofofTheorem3.1isbasedonthe followinglemmatagivenin[16]which
shows an upper bound for standardized Sinc functions.
Lemma 4.1. [16] For k,d N and d k+2, there is a constant c R+ such
k
∈ ≥ ∈
that for
c
Gk(x):=χR\[−1,1](x)π2kx2 +πk|x|kexp(−x2)
where χ (x) is characteristic function on set A, it holds
A
G Lp(R), p [1, ),
k
∈ ∀ ∈ ∞
and
d
πx
πk xk sinc G (x).
k
| | · √d ≤
(cid:12) (cid:18) (cid:19)(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
Proof. We start with the inequa(cid:12)lity (cid:12)
x [0,1],sinc(x) 1 x2,
∀ ∈ ≤ −
8
where the right-hand side term is the parabola that goes through the two ex-
treme values of sinc(x) within this interval. We will first show that
x2 d
x [0,√d],sincn x/√d 1 exp x2 . (4.1)
∀ ∈ ≤ − d ≤ −
(cid:16) (cid:17) (cid:18) (cid:19)
(cid:0) (cid:1)
For this purpose, we define the positive function
x2 d
p(x)=exp x2 1 .
− d
(cid:18) (cid:19)
(cid:0) (cid:1)
The derivative of p(x) is given by
∂p(x) 2exp(x2)x3 1 x2/d d
= − −
∂x d(1 x2/d)
− (cid:0) (cid:1)
and is always negative for x [0,√d]. Therefore, the maximum of p(x) within
∈
the interval occurs at x=0
sup p(x)=p(0)=1,
x (0,√n)
∈
which proves (4.1).
Second, for x √d, set
≥
k+2
d 2
c := max .
k n≥k+2 πd−k−2!
d
We note that πk xk sinc(πx/√d) is also bounded by
| | ·
(cid:12) (cid:12)
(cid:12) (cid:12) d c
(cid:12)πk xk sinc(cid:12) πx/√d k . (4.2)
| | · ≤ x2π2
(cid:12) (cid:16) (cid:17)(cid:12)
(cid:12) (cid:12)
The existence of c follows from(cid:12) the convergen(cid:12)ce of the sequence
k
k+2
d 2
.
πd k+2
− !
d k+2
≥
Then one follows
d k+2
√d √d 1 c
k
.
xπ! ≤ x ! · πd ≤ xk+2πk+2
We get a bound ck that is independent of d by noticing that
x2π2
d
πx √d
sincd .
(cid:12) (cid:18)√d(cid:19)(cid:12)≤ πx!
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
9
Finally, we define the function G (x), which is independent of d, by suit-
k
ably combining the right-hand sides of (4.1) and (4.2). The function πk xk
| | ·
n
sinc πx is uniformly bounded from above by an L ( ,+ ) function
√n p −∞ ∞
(cid:12) (cid:16) (cid:17)(cid:12)
(cid:12)Gk(x) wher(cid:12)e p [1,+ ).
(cid:12) (cid:12) ∈ ∞
We have all the ingredients to prove our results now.
Proof of Theorem 3.1. Set
x 12
L (x):=dln sinc . (4.3)
n
2 d
r !!
Due to the symmetry, we may assume x 0. By Taylor’s theorem, for any
≥
x [0,1] and d N, it holds
∈ ∈
x 12 x21 1
sinc =1 +O
2r d ! − 2 d (cid:18)d2(cid:19)
and
ln(1+x)=x+O(x2).
Then for any x [0,1] and d N, it holds
∈ ∈
x2 1 1 x2 1
L (x)=dln 1 +O = +O . (4.4)
n − 2 d d2 − 2 d
(cid:18) (cid:18) (cid:19)(cid:19) (cid:18) (cid:19)
Combing (4.3) and (4.4), we have
x 12 x2 1
sincd =exp 1+O . (4.5)
2r d ! (cid:18)− 2 (cid:19)(cid:18) (cid:18)d(cid:19)(cid:19)
Furthermore, it holds B (ω) =sincd ω , and B Cd 1(R). The later yields
d∧ 2 d ∈ −
for k d 1: (cid:0) (cid:1)
≤ −
ω
B(k) ∧(ω)=ikωksincd .
d 2
h i (cid:16) (cid:17)
Consequently, one obtains
k+1
d 2 B(k) d x+ d ∧(ω) = ikωksincd ω 12
12 d 12 2 2 d
(cid:18) (cid:19) " r !# r !
ω2 1
= ikωkexp 1+O
− 2 d
(cid:18) (cid:19)(cid:18) (cid:18) (cid:19)(cid:19)
1 x2 1 ∧
= Dk exp 1+O (ω)
√2π − 2 · d
(cid:20) (cid:26) (cid:18) (cid:19) (cid:18) (cid:18) (cid:19)(cid:19)(cid:27)(cid:21)
10