On the First Kind r-Whitney Numbers and the Combinatorics of 0-1 Tableaux

On the First Kind r-Whitney Numbers and the

Combinatorics of 0-1 Tableaux

Roberto B. Corcino and Cristina B. Corcino

Cebu Normal University, Cebu City, Philippines 6000

Date Submitted: September 30, 2015 Originality: 91%

Date Revised: December 18, 2016 Plagiarism Detection: Passed

ABSTRACT

The limit expression, denoted by 𝐹𝛼,𝛾(𝑛, 𝑘),

𝐹

(𝑛, 𝑘) = lim

𝛽→0

[Δ

(𝛽𝑡 + 𝛾|𝛼)

]

𝑘! 𝛽

was evaluated completely that yields an explicit formula in elementary symmetric function form. The limit 𝐹𝛼,𝛾(𝑛, 𝑘) has possessed some necessary properties like recurrence relations, generating functions, orthogonality relation, inverse relation and matrix factorization. It has been identified that the numbers 𝐹𝛼,𝛾(𝑛, 𝑘) are certain generalization of Gould-Hopper numbers, which are useful in biology and reliability theory. One can easily verify that these numbers are equivalent to the first kind r-Whitney numbers using their existing properties. In this present paper, a combinatorial meaning for r -Whitney numbers of the first kind in the context of (0, 1)-tableau is constructed using the explicit formula in symmetric function form. Moreover, some convolution-type formulas are derived that, consequently, yield LU -factorization of the matrix whose entries are the first kind r-Whitney numbers.

Keywords: Stirling numbers, generating function, recurrence relation, generalized factorial, 0-1 tableau, convolution identity, difference operator, r-Whitney numbers.

INTRODUCTION

The generalized factorial, denoted by

(𝛽𝑡 + 𝛾|𝛼)_𝑛, is defined by

(𝛽𝑡 + 𝛾|𝛼)𝑛= ∏ 𝛽𝑡 + 𝛾 − 𝑘𝛼 𝑛−1

𝑘=0

.

Using the well-known identity of the kth difference operator that gives

[𝛥𝑡𝑘(𝛽𝑡 + 𝛾|𝛼)𝑛]_𝒕=𝟎

= ∑(−1)𝑘−𝑗(𝑘 𝑗)

𝑘

𝑗=0

(𝛽𝑗 + 𝛾|𝛼)𝑛,

the limit

𝐹𝛼,𝛾(𝑛, 𝑘) = lim 𝛽→0

[Δ𝑡𝑘(𝛽𝑡 + 𝛾|𝛼)𝑛]_𝑡=0

𝑘! 𝛽𝑘

has been evaluated completely in (Corcino et al., 2005). More precisely, the limit yields the following explicit formula in elementary symmetric function form

𝐹𝛼,𝛾(𝑛, 𝑘) = ∑ ∏(𝛾 − 𝑗𝑞𝛼) (1) 𝑛−𝑘

𝑞=1 0≤𝑗1<⋯<𝑗𝑛−𝑘≤𝑛−1

When 𝛼 = 1, the above kth difference operator was considered by Gould and Hopper (1962) and was called Gould-Hopper numbers. Charalambides (1981) obtained certain occupancy distribution in terms of these numbers which is useful in biology and reliability theory, and together with Koutras, Charalambides (1983) made a thorough investigation of these numbers and obtained several interesting properties which, consequently, give some combinatorial and statistical applications.

The limit 𝐹𝛼,𝛾(𝑛, 𝑘) has been identified as certain generalization of Stirling numbers of the first kind. In fact, when 𝛼 = 1 and 𝛾 = 0, the explicit formula in (1) yields

𝐹1,0(𝑛, 𝑘) = (−1)𝑛−𝑘 ∑ 𝑗1𝑗1… 𝑗𝑛−𝑘 0≤𝑗1<⋯<𝑗𝑛−𝑘≤𝑛−1

which is exactly the explicit formula in symmetric function form for the first kind Stirling numbers. Some necessary properties of 𝐹𝛼,𝛾(𝑛, 𝑘) parallel to those of the first kind Stirling numbers have been derived including the three types of recursive formula

1) 𝐹𝛼,𝛾(𝑛 + 1, 𝑘) = 𝐹𝛼,𝛾(𝑛, 𝑘 − 1) + (𝛾 − 𝑛𝛼)𝐹𝛼,𝛾(𝑛, 𝑘)

2) 𝐹𝛼,𝛾(𝑛 + 1, 𝑘 + 1) = ∑

(𝛾|𝛼)𝑛+1

(𝛾|𝛼)𝑗+1

𝑛

𝑗=𝑘

𝐹𝛼,𝛾(𝑛, 𝑘)

3) 𝐹𝛼,𝛾(𝑛, 𝑘) = ∑(−1)𝑗 𝑛−𝑘

𝑗=0

(𝛾 − 𝑛𝛼)𝑗 _𝐹

𝛼,𝛾(𝑛 + 1, 𝑘 + 𝑗 + 1)

the horizontal generating function,

∑ 𝐹𝛼,𝛾(𝑛, 𝑘)𝑡𝑘= (𝑡 + 𝛾|𝛼)𝑛

𝑛

𝑘=0

and the exponential generating function

∑ 𝐹𝛼,𝛾(𝑛, 𝑘)

𝑡𝑛

𝑛!

𝑛≥0

= 1

𝑘! (1 + 𝛼𝑡)

𝛾

𝛼 _{[log(1 + 𝛼𝑡)} 1 𝛼_]

𝑘

.

(Corcino et al., 2005) Certain inverse matrix relation involving 𝐹𝛼,𝛾(𝑛, 𝑘) has been established using the orthogonality relation

∑ 𝐹_𝛽,−𝑟(𝑚, 𝑘) {𝑘 𝑛}𝛽,𝑟

= 𝑚

𝑘=𝑛

∑ {𝑚

𝑘}𝛽,𝑟 𝐹𝛽,−𝑟(𝑘, 𝑛) = 𝛿𝑚𝑛 (𝑚 𝑚

𝑘=𝑛 ≥ 𝑛)

where {𝑘

𝑛}𝛽,𝑟 denotes the (𝑟, 𝛽)-Stirling numbers and 𝛿𝑚𝑛 is the Kronecker delta defined by 𝛿𝑚𝑛= 1 if 𝑚 = 𝑛, and 𝛿𝑚𝑛 =

0 if 𝑚 ≠ 𝑛. Consequently, an inverse relation is obtained

𝑓𝑛= ∑ {

𝑛 𝑘}

𝑛

𝑘=0 𝛽,𝑟

𝑔𝑘 ⟺ 𝑔𝑛

= ∑ 𝐹𝛽,−𝑟 (𝑛, 𝑘) 𝑓𝑘

𝑛

𝑘=0

which is parallel to the well-known binomial inversion formula

𝑓𝑛= ∑ (

𝑛 𝑘)

𝑛

𝑘=0

𝑔𝑘⟺ 𝑔𝑛= ∑ (−1)𝑛−𝑘 (

𝑛 𝑘) 𝑓𝑘

𝑛

𝑘=0

(Corcino and Herrera, 2005). With these properties, one can easily verify that

𝐹

(𝑛, 𝑘) = 𝑤

(𝑛, 𝑘)

where 𝑤𝛼,𝛾(𝑛, 𝑘) are the first kind r -Whitney numbers of Mező (2010). Thus, equation (1) can be written as

= ∑ ∏(𝛾

𝑛−𝑘

𝑞=1 0≤𝑗1<⋯<𝑗𝑛−𝑘≤𝑛−1

− 𝑗𝑞𝛼). (2)

The distribution of r-Whitney numbers of the first kind 𝑤𝛼,−𝛾(𝑛, 𝑘) has been shown to be unimodal and asymptotically normal by Corcino-Corcino (2011) in which the maximum occurs at the index 𝑘 = 𝐾𝛼,𝛾,(𝑛) where

𝐾𝛼,𝛾(𝑛) = [− 1 𝛼log (

[𝛾 + 1_−𝛼 ] + 𝑛 − 1 [𝛾 + 1_−𝛼 ] − 1

) + 𝜊(1)].

It is worth-mentioning that these numbers have been given combinatorial interpretation by Cheon and Jung (2012) in terms of Dowling lattices and further investigated by Merca (2014), Corcino et. al (2014) and Mező (2015).

In this present paper, we give a certain combinatorial meaning of

𝑤𝛼,−𝛾(𝑛, 𝑘) which, consequently, yields some identities including other forms of convolution-type identities.

COMBINATORIAL INTERPRETATION

Consider a partition of integer t

𝑆 = (𝑠

≥ 𝑠

≥ ⋯ ≥ 𝑠

)

and the Ferrers diagram of shape S whose cells are filled up with 0 and 1 such that only one 1 is present in each column. If the ‘filling’ is denoted by 𝑔 = (𝑔𝑖𝑗)_{1≤𝑗≤𝑠}

𝑖 then the pair 𝜑 = (𝑆, 𝑔) is called a (0 – 1)-tableau. (De Medicis and Leroux, 1995)

On the other hand, suppose that the lengths of the columns of a Ferrers diagram belong to the sequence of nonnegative

integers 𝐴 = (𝛼𝑖)𝑖≥0 such that the order of the length is decreasing. Then the list of

these columns is called an

𝐴

-

tableau

.

(De Medicis and Leroux, 1995)

If

Ψ

_{(𝑙, 𝑝)}

_{is the collection of}

A

– tableaux containing exactly

p

different columns such that the lengths

belong to the set

𝐴 = {𝛼

, 𝛼

, … , 𝛼

},

then

|Ψ

_{(𝑙, 𝑝)| = (}

)

Now, suppose that, for a given A – tableau in Ψ𝐴_{(𝑙, 𝑝), each column c into a} column of length 𝜔(|𝒸|). Then is transformed a new tableau which is called

𝐴𝜔− 𝑡𝑎𝑏𝑙𝑒𝑎𝑢 is obtained. If 𝜔(|𝒸|)=|𝒸|, then the 𝐴𝜔−tableau is simply the A – tableau. Now, 𝐴𝜔∗ − 𝑡𝑎𝑏𝑙𝑒𝑎𝑢 is defined as a (0 − 1) −tableau which is constructed by filling up the cells of an 𝐴𝜔−tableau with 0 and 1 such that each column must contain exactly one 1. The notation Ψ𝐴𝜔∗_{(𝑙, 𝑝) is} used for the collection of all such

𝐴∗𝜔−tableaux.

It can easily be seen that every

(𝑛 − 𝑘) −combination 𝑗1𝑗2… 𝑗𝑛−𝑘 of the set (𝑖)𝑖=0𝑛−1 can be presented geometrically by an element 𝜙 in Ψ𝐴(𝑛 − 1, 𝑛 − 𝑘) with

𝑗𝑖 as the length of (𝑛 − 𝑘 − 𝑖 + 1)𝑡ℎ column of 𝜙 where 𝐴 = (𝑖)_𝑖=0𝑛−1. Hence, with 𝜔(|𝒸|) = 𝛾 − |𝒸|𝛼, equation (2) may be written as

𝑤

(𝑛, 𝑘)

= ∑ ∏ 𝜔(|𝒸|). (3)

Thus, using (3), we have the following result is obtained.

Theorem 1. The number of 𝐴𝜔∗ - tableaux in Ψ𝐴𝜔_{(𝑛 − 1, 𝑛 − 𝑘) where 𝐴 =} {0,1,2, … , 𝑛 − 1} such that 𝜔(|𝒸|) = 𝛾 − |𝒸|𝛼 is equal to 𝑤𝛼,−𝛾(𝑛, 𝑘).

Remark 1. By making use of (1) and Theorem 1 with 𝛼 = −1 and 𝛾 = 0, the signless first kind Stirling numbers

|𝑠(𝑛, 𝑘)| equal the number of A – tableaux inΨ𝐴𝜔∗_{(𝑛 − 1, 𝑛 − 𝑘) where 𝐴 = (𝑖)}

𝑖=0 𝑛−1_.

Remark 2. The first kind 𝑟-Stirling numbers [𝑛 + 𝑟

𝑘 + 𝑟]𝑟can also interpreted using (1) and Theorem 1 with 𝛼 = −1 and

𝛾 = 𝑟 (a nonnegative integer) as number of the ways to construct an 𝐴𝜔∗ − 𝑡𝑎𝑏𝑙𝑒𝑎𝑢 in

Ψ𝐴𝜔∗_{(𝑛 − 1, 𝑛 − 𝑘) where 𝐴 = (𝑖)}

𝑖=0 𝑛−1 such that 𝜔(|𝑐|) = |𝑐| + 𝑟.

Remark 3. With the appropriate choice of the values of 𝛼 and 𝛾, the other Stirling type numbers of the first kind can also be interpreted in terms of 0-1 tableau by making use of Theorem 1.

THE COMBINATORICS OF

A-TABLEUAX

Suppose that 𝜌 is an A – tableau in

Ψ𝐴_{(𝑛 − 1, 𝑛 − 𝑘) with 𝐴 = (𝑖)} 𝑖=0 𝑛−1_{, and}

𝜔𝐴(𝜌) = ∏ 𝜔(|𝑐|)

𝑐∈𝜌

= ∏(𝛾 − 𝑗𝑖𝛼) , 𝑗𝑖

𝑛−𝑘

𝑖=1

∈ (𝑖)𝑖=0𝑛−1.

If 𝛾 = 𝛾1+ 𝛾2

for some

𝛾

and

𝛾

,

then,

with

𝜔𝐴(𝜌) = ∏(𝛾1+ 𝜔∗(𝑗𝑖)) 𝑛−𝑘

𝑖=1

= ∑ 𝛾₁𝑛−𝑘−𝑟 ∑ ∏ 𝜔∗_(𝑞

𝑖). 𝑟

𝑖=1 𝑗𝑖≤𝑞1<𝑞2<⋯<𝑞𝑟≤𝑗𝑛−𝑘

𝑛−𝑘

𝑟=0

Suppose 𝐵𝜌 is the set of all A – tableaux corresponding to 𝜌 such that for each 𝜓 ∈ 𝐵𝜌 either

𝜓 has no column whose weight is 𝛾1, or

𝜓 has one column whose weight is 𝛾1, or

𝜓 has (n-k) columns whose weight are

𝛾1.

Then, we may write

𝜔𝐴(𝜌) = ∑ 𝜔𝐴(𝜓) 𝜓∈ 𝐵_𝜌

.

Now, if

r

columns in

𝜓

have weights

other than

𝛾

, then

𝜔𝐴(𝜓) = 𝛾1𝑛−𝑘−𝑟∏ 𝜔∗(𝑞𝑖) 𝑟

𝑖=1

,

where

𝑞

, 𝑞

, … , 𝑞

∈ {𝑗

∶ 𝑖 =

1, … , 𝑛 − 𝑘 }.

Hence, equation (3) may

be written as

𝑤𝛼,−𝛾(𝑛, 𝑘) = ∑ ∑ 𝜔𝐴(𝜓).

𝜓∈𝐵𝜌

𝜌∈Ψ𝐴(𝑛−1,𝑛−𝑘)

Note that for each r, there correspond (𝑛−𝑘_𝑟 ) tableaux with r distinct columns having weights 𝜔∗(𝑞𝑖) , 𝑞𝑖 ∈

{𝑗𝑖 ∶ 𝑖 = 1, … , 𝑛 − 𝑘 }. Note that Ψ𝐴(𝑛 −

1, 𝑛 − 𝑘) has (𝑛_𝑘) elements. Hence, for each 𝜌 ∈ Ψ𝐴(𝑛 − 1, 𝑛 − 𝑘), there are

distinct that belong to 𝐵𝜌. Hence, every distinct tableau 𝜓 appears

(𝑛_𝑘)(𝑛−𝑘_𝑟 ) (𝑛_𝑟) = (

𝑛 − 𝑟

𝑘 )

times in the collection. Consequently, 𝑤𝛼,−𝛾(𝑛, 𝑘) = ∑ (

𝑛 − 𝑟 𝑘 ) 𝛾1

𝑛−𝑘−𝑟 _{∑ ∏ 𝜔}∗_(|𝑐|)

𝑐∈𝜓 𝜓∈𝐵𝑟

𝑛−𝑘

𝑟=0

Can be obtained; where 𝐵𝑟 is the collection of all tableaux 𝜓 having r distinct columns with lengths belong to(𝑖)_𝑖=0𝑛−1. Reindexing the double sum, we get

𝑤𝛼,−𝛾(𝑛, 𝑘) = ∑ ( 𝑗 𝑘) 𝛾1

𝑗−𝑘

∑ ∏ 𝜔∗_(|𝑐|) 𝑐∈𝜓∗

𝜓∗∈𝐵𝑛−𝑗

𝑛

𝑗=𝑘

.

Clearly, 𝐵𝑛−𝑗 = Ψ𝐴(𝑛 − 1, 𝑛 − 𝑗). Thus, equation (5) yields the following result.

Theorem 2. The numbers 𝑤𝛼,−𝛾(𝑛, 𝑘) satisfy the following identity

𝑤𝛼,−𝛾(𝑛, 𝑘) = ∑ (

𝑗

𝑘)

𝑛

𝑗=𝑘

𝛾₁𝑗−𝑘 𝑤𝛼,−𝛾2(𝑛, 𝑗)

where 𝛾 = 𝛾1+ 𝛾2 for some numbers 𝛾1

𝑎𝑛𝑑 𝛾2

As a direct consequence of Theorem 6, the first kind r-Stirling numbers as well as the first kind non-central Stirling numbers in terms of the classical first kind Stirling numbers can be expressed. First, when 𝛼 = −1, 𝛾1 = 𝑟 and

𝛾2= 0, Theorem 6 yields

𝑤−1,−𝑟(𝑛, 𝑘) = ∑ (

𝑗 𝑘)

𝑛

𝑗=𝑘

𝑟𝑗−𝑘 _𝑤

1,0(𝑛, 𝑗).

Using equations in Remark 1, the following corollary is obtained.

Corollary 3. The first kind r-Stirling

numbers [𝑛 + 𝑟

𝑘 + 𝑟]𝑟equal

[𝑛 + 𝑟

𝑘 + 𝑟]𝑟= ∑ (

𝑗 𝑘)

𝑛

𝑗=𝑘

𝑟𝑗−𝑘 _{|𝑠(𝑛, 𝑗)|}

Now, when 𝛼 = 1, the non-central first kind Stirling numbers 𝑠(𝑎)(𝑛, 𝑘) can be written as

𝑠_(𝑎)(𝑛, 𝑘) = 𝑤1,−𝑎(𝑛, 𝑘)

(Koutras, 1982). Hence, with 𝛾2 = 0 and

𝛾1= 𝑎, Theorem 6 gives the following corollary using the first equation in Remark 1.

Corollary 4. The first kind non-central

Stirling numbers denoted by 𝑠_(𝑎)(𝑛, 𝑘) is given by

𝑠(𝑎)(𝑛, 𝑘) = ∑ (

𝑗

𝑘)

𝑛

𝑗=𝑘

𝑎𝑗−𝑘_{𝑠(𝑛, 𝑗).}

CONVOLUTION-TYPE IDENTITIES

Theorem 5 (Corcino-Herrera, 2009). The first kind r-Whitney numbers satisfy the following convolution-type identity

(𝑘 𝑘1

) 𝑤𝛼,−𝛾(𝑛, 𝑘)

= ∑ 𝑤𝛼1,−𝛾1(𝑚, 𝑘1)

𝑛

𝑚=0

𝑤𝛼2,−𝛾2(𝑛 − 𝑚, 𝑘2)

where𝑘 = 𝑘1+ 𝑘2and𝛾 = 𝛾1+ 𝛾2.

This identity can be rewritten as

(𝑘1+ 𝑘2)! 𝑤𝛼,−𝛾(𝑛, 𝑘1+ 𝑘2)

= ∑ 𝑘1! 𝑤𝛼1,−𝛾1(𝑚, 𝑘1)

𝑛

𝑚=0

𝑘2! 𝑤𝛼2,−𝛾2(𝑛

− 𝑚, 𝑘2). (4)

The following matrix factorization immediately follows from (4).

Corollary 6. The following

LU-factorization of the matrix

((𝑖 +

𝑗)! 𝑤

(𝑛, 𝑖 + 𝑗))

((𝑖 + 𝑗)! 𝑤𝛼,−𝛾(𝑛, 𝑖 + 𝑗))0≤𝑖,𝑗≤𝑛

= (𝑖! 𝑤𝛼1,−𝛾1(𝑚, 𝑖))_{0≤𝑚,𝑖≤𝑛}

𝑇

(𝑗! 𝑤𝛼2,−𝛾2(𝑛 − 𝑚, 𝑗))

0≤𝑚,𝑗≤𝑛.

The next theorem contains another form of convolution-type formula for

𝐹𝛼,𝛾(𝑛, 𝑘) which will be proved using the combinatorics of A-tableau.

Theorem 7. The numbers 𝐹𝛼,𝛾(𝑛, 𝑘) have convolution – type formula

𝑤𝛼,−𝛾(𝑚 + 𝑗, 𝑛)

= ∑𝑤𝛼,−𝛾(𝑚, 𝑘) 𝑤𝛼,−𝛾+𝑚𝛼(𝑗, 𝑛 − 𝑘). 𝑛

𝑘=0

Proof. Suppose that 𝜙1 is a tableau with exactly 𝑚 − 𝑘 distinct columns with lengths belong to Ω1 = (𝑖)𝑖=0𝑚−1 and 𝜙2 is a tableau with exactly 𝑗 − 𝑛 + 𝑘 distinct

columns whose lengths are in the set

Ω2= (𝑖)𝑖=𝑚 𝑚+𝑗−1

. Then 𝜙1∈ ΨΩ1(𝑚 −

1, 𝑚 − 𝑘) and 𝜙2∈ ΨΩ2(𝑗 − 1, 𝑗 − 𝑛 +

𝑘). Notice that by joining the columns of

𝜙1 and 𝜙2, we obtain an A – tableau 𝜙 with 𝑚 + 𝑗 − 𝑛 distinct columns whose lengths are in the set 𝐴2= {0,1,2, … , 𝑚 +

𝑗 − 1}, that is, 𝜙 ∈ ΨΩ1_{(𝑚 + 𝑗 − 1, 𝑚 +} 𝑗 − 𝑛). Hence,

∑ 𝜔𝐴(𝜙) 𝜙∈Ψ𝐴(𝑚+𝑗−1,𝑚+𝑗−𝑛)

= ∑ { ∑ 𝜔𝐴1(𝜙1)

𝜙1∈ΨΩ1(𝑚−1,𝑚−𝑘)

}

𝑛

𝑘=0

×

× { ∑ 𝜔𝐴2(𝜙2)

𝜙2∈ΨΩ2(𝑗−1,𝑗−𝑛+𝑘)

}.

Note that

∑ 𝜔𝐴2(𝜙2)

𝜙2∈ΨΩ2(𝑗−1,𝑗−𝑛+𝑘)

= ∑ ∏ (𝛾 − 𝑔𝑞𝛼)

𝑗−𝑛+𝑘

𝑞=1 𝑚≤𝑔1<𝑔2<⋯<𝑔𝑗−𝑛+𝑘≤𝑚+𝑗−1

= ∑ ∏ (𝛾 − (𝑚 + 𝑔𝑞)𝛼) 𝑗−𝑛+𝑘

𝑞=1 0≤𝑔1<𝑔2<⋯<𝑔𝑗−𝑛+𝑘≤𝑗−1

= 𝑤𝛼,−𝛾+𝑚𝛼(𝑗, 𝑛 − 𝑘).

Also, using equation (3), we have

∑ 𝜔𝐴1(𝜙1) = 𝑤𝛼,−𝛾(𝑚, 𝑘) ,

𝜙1∈ΨΩ1(𝑚−1,𝑚−𝑘)

∑ 𝜔𝐴(𝜙) 𝜙∈Ψ𝐴_{(𝑚+𝑗−1,𝑚+𝑗−𝑛)}

= 𝑤𝛼,−𝛾(𝑚 + 𝑗, 𝑛)

Thus,

𝑤𝛼,−𝛾(𝑚 + 𝑗, 𝑛)

= ∑ 𝑤𝛼,−𝛾(𝑚, 𝑘)𝑤𝛼,−𝛾+𝑚𝛼(𝑗, 𝑛 − 𝑘). 𝑛

The following corollary contains another LU-factorization of the matrix whose entries are the first kind r-Whitney numbers.

Corollary 8. The following LU-factorization of the matrix (𝑤𝛼,−𝛾(𝑖 +

𝑗, 𝑛))

0≤𝑖,𝑗≤𝑛 holds

(𝑤𝛼,−𝛾(𝑖 + 𝑗, 𝑛)) 0≤𝑖,𝑗≤𝑛

= (𝑤𝛼,−𝛾(𝑖, 𝑘))

0≤𝑖,𝑗≤𝑛(𝑤𝛼,−𝛾+𝑖𝛼(𝑗, 𝑛

− 𝑘))

0≤𝑗,𝑘≤𝑛 𝑇

.

The following theorem gives another form of convolution-type formula.

Theorem 9. The numbers 𝑤𝛼,−𝛾(𝑛, 𝑘) satisfy the second form of convolution-type formula

𝑤𝛼,−𝛾(𝑛 + 1, 𝑚 + 𝑗 + 1)

= ∑ 𝑤𝛼,−𝛾(𝑘, 𝑚)𝑤𝛼,−𝛾+(𝑘+1)𝛼(𝑛 𝑛

𝑘=0

− 𝑘, 𝑗).

Proof. Let 𝜙1 be a tableau with 𝑘 − 𝑚 columns whose lengths are in 𝐴1=

{0,1, … , 𝑘 − 1}, and 𝜙2 be a tableau with

𝑛 − 𝑘 − 𝑗 columns whose lengths are in

𝐴2 = {𝑘 + 1, … , 𝑛}. Then 𝜙1∈ 𝑇𝑑𝐴1(𝑘 −

1, 𝑘 − 𝑚), 𝜙2∈ 𝑇𝑑𝐴2(𝑛 − 𝑘 − 1, 𝑛 − 𝑘 −

𝑗).

Using the same above argument, the convolution formula can be derived easily.

Corollary 10. The following LU-factorization of the matrix (𝑤𝛼,−𝛾(𝑛 +

1, 𝑖 + 𝑗 + 1))

0≤𝑖,𝑗≤𝑛 holds (𝑤𝛼,−𝛾(𝑛 + 1, 𝑖 + 𝑗 + 1))

0≤𝑖,𝑗≤𝑛

= (𝑤𝛼,−𝛾(𝑘, 𝑖)) 0≤𝑘,𝑖≤𝑛 𝑇

(𝑤𝛼,−𝛾+(𝑘+1)𝛼(𝑛

− 𝑘, 𝑗))

0≤𝑘,𝑗≤𝑛.

Remark 4. When 𝛼 = −1 and 𝛾 = 0, Theorem 7 and 8 give

|𝑠(𝑚 + 𝑗, 𝑛)| = ∑|𝑠(𝑚, 𝑘)| [ 𝑗 + 𝑚

𝑛 − 𝑘 + 𝑚]

𝑛

𝑘=0 _𝑚

|𝑠(𝑛 + 1, 𝑚 + 𝑗 + 1)|

= ∑|𝑠(𝑘, 𝑚)| [_{𝑗 + 𝑘 + 1}𝑛 + 1 ]

𝑛

𝑘=0 𝑘+1

.

Remark 5. When 𝛼 = −1 and 𝛾 = 𝑟 any nonnegative integer, Theorem 7 and 8 yield

[𝑚 + 𝑗 + 𝑟

𝑛 + 𝑟 ]𝑟= ∑ [

𝑚 + 𝑟 𝑘 + 𝑟]

𝑛

𝑘=0 _𝑟

[ 𝑗 + 𝑟

𝑛 − 𝑘 + 𝑟 + 1]𝑟+𝑚

[ 𝑛 + 𝑟 + 1

𝑚 + 𝑗 + 𝑟 + 1]_𝑟= ∑ [ 𝑘 + 𝑟 𝑚 + 𝑟] 𝑛

𝑘=0 𝑟

[ 𝑛 + 𝑟 + 1 𝑗 + 𝑘 + 𝑟 + 1]_𝑟+𝑘+1

Remark 6.

Moreover, when

𝛼 = 1

,

Theorem 7 and 8 yield

𝑠(𝛾)(𝑚 + 𝑗, 𝑛) = ∑ 𝑠(𝛾)(𝑚, 𝑘)𝑠(𝛾−𝑚)(𝑗, 𝑛 − 𝑘) 𝑛

𝑘=0

𝑠(𝛾)(𝑛 + 1, 𝑚 + 𝑗 + 1)

= ∑ 𝑠(𝛾)(𝑘, 𝑚)𝑠(𝛾−𝑘−1)(𝑛 − 𝑘, 𝑗). 𝑛

𝑘=0

It is interesting to have these identities since they are not known in the literature of the classical Stirling numbers, first kind r-Stirling numbers, and first kind non-central Stirling numbers and that these identities further give LU-factorizations of matrices whose entries are above-mentioned numbers.

RECOMMENDATION

A q-analogue of the first kind Stirling numbers was first defined and explored by Carlitz (1948). A follow up study was done by Gould (1968) giving further investigation of this q-analogue.

attracted to work on them resulting to possession of numerous literatures related to these numbers. For instance, Milne (1982) gave a combinatorial meaning of q-Stirling numbers of the first kind by means of restricted growth function, while Wachs and White (1991) gave an interpretation in terms of set partition statistics.

For possible future research work, one may try to define and investigate a q -analogue of the first kind r-Whitney numbers parallel to the work of Carlitz (1948) and Gould (1968).

REFERENCES

Andre Z. Broder. (1984), The r-stirling numbers, Discrete Math. 49, pp. 241-259.

David Branson. (2000), Stirling Numbers and Bell numbers: Their Role in Combinatorics and Probability, The Mathematical Scientist, 25, pp. 1-31.

Leonard Carlitz. (1948), q-Bernoulli Numbers and Polynomials, Duke Math . J. 15, pp. 987-1000.

Leonard Carlitz. (1980), Weighted Stirling Numbers of the first and the second kind- I, The Fibonacci Quarterly, 18, pp. 146-163.

Charalambos A. Charalambides. (1984), On Weighted Stirling and Other Related Numbers and Some Combinatorial applications, Fibonacci Quart., 22, pp. 296-309.

Charalambos .A. Charalambides and Markos Koutras (1983), On the Differences of

Generalized Factorials at an Arbitrary Point and Their Combinatorial Applications, Discrete Math., 47, pp. 183-201.

Charalambos A. Charalambides and Jitender Singh. (1988), A review of the Stirling numbers, their generalization and Statistical Applications, Commun. Statist.-Theory Meth., 20(8), pp. 2533-2595.

Louis Comtet (1974), Advanced Combinatorics, Reidel, Dordrecht, The Netherlands.

Gi-Sang Cheon and Ji-Hwan Jung (2012), r -Whitney Number of Dowling Lattices,Discrete Math., 312, pp. 2337-2348.

Roberto B. Corcino and Cristina B. Corcino. (2011), On the Maximum of Generalized Stirling Numbers, Utilitas Mathematica, 86, pp. 241-256.

Roberto B. Corcino, Cristina B. Corcino and Miraluna L. Herrera. (2005), On the Limiting Form of the Differences of the Generalized Factorial, Journal of Research for Science, Computing and Engineering, 2(3), pp. 1-7.

Roberto B. Corcino and Miraluna L. Herrera. (2009), Some Properties of the Limit of the Differences of the Generalized Factorial, Matimyas Mathematika, 32(2), pp. 21-28.

Roberto B. Corcino and Miraluna L. Herrera (2005), The Orthogonality and Inverse Relations of the Limit of the Differences of the Generalized Factorial, Mindanao Forum, 19(1), pp. 117-124.

Convolution-Type identities, Matimyas Mathematika, 23, pp. 21-29.

Roberto B. Corcino, Maribeth B. Montero and Susan Ballenas. (2014), Schlömilch-Type Formula for r-Whitney Numbers of the First Kind, Matimyas Matematika, 37(1-2), pp. 1-10.

Henry W. Gould. (1968), The q-Stirling Numbers of the First and Second Kind, Duke Math. J.28, pp. 281-

289.

Anne De Medicis and Pierre Leroux. (1995),Generalized Stirling Numbers, Convolution Formulas and p, q -Analogues, Can. J. Math. 47(3), pp. 464-499.

S.C. Milne. (1982), Restricted growth

functions, rank row matchings of partition lattices, and q-Stirling

numbers, Adv. Math.43, pp. 173–196.

Henry W. Gould and A.T. Hopper (1962), Operational Formulas Connected with Two Generalizations of Hermite Polynomials, Duke Math. J., 29, (1962), 51-63.

Leetsch Charlse Hsu and Peter J-S. Shiue (1998), A unified approach to generalized Stirlingnumbers, Advances in Appl. Math., 20, pp. 366-384.

Markos Koutras (1982), Non-Central Stirling Numbers and some

Applications, Discrete Math.,42, pp. 73-79.

Mircea Merca (2014), A new connection between r-Whitney numbers and Bernoulli polynomials, Integral transform Spec. Funct.

25(12), DOI:10.

1080/10652469.2014.940580.

Istvan Mező. (2010), A new formula for the Bernoulli polynomials, Result. Math., 58(3), pp. 329-335.

Istvan Mező and José L. Ramírez (2015), The Linear Algebra of the r-Whitney Matrices, Integral Transforms Spec. Funct., 26(3), pp. 213-225.

John Riordan. (1958), An Introduction to Combinatorial Analysis, John Wiley and

Son, Inc.

Richard P. Stanley. (1986), Enumerative Combinatorics, Wadsworth and Brooks/Cole, Monterey.

M. Wachs, D. White (1991), p, q-Stirling numbers and partition statistics, J. Combin. Theory Ser. A, 56, pp. 27–46.