Additionally, for some particular cases we study their relationship witht- successive associated Stirling numbers and theirq-analogue

Miskolc Mathematical Notes HU e-ISSN 1787-2413 Vol. 20 (2019), No. 1, pp. 99–114 DOI: 10.18514/MMN.2019.2536

THE GENERALIZED t-COMTET NUMBERS AND SOME COMBINATORIAL APPLICATIONS

HAC `ENE BELBACHIR, CAROLINA FORERO, AND JOS ´E L. RAM´IREZ Received 22 February, 2018

Abstract. In the present article we use a combinatorial approach to generalize the Comtet num- bers. In particular, we establish some combinatorial identities, recurrence relations and gen- erating functions. Additionally, for some particular cases we study their relationship witht- successive associated Stirling numbers and theirq-analogue.

2010Mathematics Subject Classification: 11B83; 11B73; 05A15; 05A19

Keywords: Stirling numbers of the second kind,t-successive associated Stirling numbers, Comtet numbers, combinatorial identities

1. INTRODUCTION

It is well-known that the Stirling numbers of the second kind˚n

k count the number of partitions of a set withnelements intoknon-empty blocks. This sequence satisfies the recurrence relation

(n k )

Dk (n 1

k )

C (n 1

k 1 )

; with the initial conditions˚₀

0 D1and˚_n

0 D˚₀

n D0.

The Stirling numbers˚_n

k can be generalized to theassociated Stirling numbers of the second kind ˚n

k m (cf. [1,4,7,8,10,11,18,20])) by means of a restriction on the size of the blocks. In particular, this sequence gives the number of partitions ofnelements intokblocks, such that each block contains at leastmelements. It is clear that˚_n

k 1D˚_n

k . This combinatorial sequence has been applied to the study of some special polynomials such as generalized Bernoulli and Cauchy polynomials, (see, e.g., [12–16]).

Recently, Belbachir and Tebtoub [2] considered a variation for the associated Stirl- ing numbers. They introduced the2-successive associated Stirling numbers of the second kind˚n

k

Œ2. This new sequence counts the number of partitions ofnelements

The research of Jos´e L. Ram´ırez was partially supported by Universidad Nacional de Colombia, Project No. 37805.

c 2019 Miskolc University Press

intokblocks, with the additional condition that each block contains at least two con- secutive elements. Moreover, the last elementn must either form a block with its predecessor or belong to another block satisfying the previous conditions. In [2], the authors derived the recurrence

(n k

)Œ2

Dk (n 1

k )Œ2

C (n 2

k 1 )Œ2

; n2k;

with the initial conditions˚0 0

Œ2D1,˚ n n 1

Œ2D0and˚n

0 D0forn1.

Inspired by these results, in this paper we aim to investigate the sequence fa^{Œt }.n; k/gn;k0, defined by the recurrence relation

a^{Œt }.n; k/Duka^{Œt }.n 1; k/Ca^{Œt }.n t; k 1/; nt k; (1.1) with the initial conditionsa^{Œt }.0; 0/D1; a^{Œt }.n; n `/D0for`D1; 2; : : : ; t 1and a^{Œt }.n; 0/D0, forn1. Moreover,fungis a sequence of real numbers.

We will call the sequence fa^{Œt }.n; k/gn;k0 the generalized t-Comtet numbers.

The reason for this name is that fortD1we recover the Comtet numbers (see, e.g., [9,21]). Note that if uk Dk, then a^{Œt }.n; k/D˚_n

k

Œt . This sequence is called by Belbachir and Tebtoub [3] as the t-successive associated Stirling numbers. IftD2 andu_k Dk, thena^Œ2.n; k/D˚n

k

Œ2. IftD1andu_k Dk, thena^Œ1.n; k/D˚n k . In this paper our goal is to give the recurrence relation, the generating function and some combinatorial identities. For some particular cases, we give combinatorial interpretations.

2. BASIC PROPERTIES

From the recurrence relation (1.1) we obtain the following generating function.

Theorem 1. Fork1, A^{Œt }_k .x/WD X

nt k

a^{Œt }.n; k/xⁿD x^{t k}

.1 u0x/.1 u1x/.1 u2x/ .1 u_kx/; (2.1) withA^{Œt }₀ .x/D_{1 u}¹_0x.

Proof. Multiplying both sides of (1.1) byxⁿand summing overnt k, we have A^{Œt }_k .x/Du_k X

nt k

a^{Œt }.n 1; k/xⁿC X

nt k

a^{Œt }.n t; k 1/xⁿ Dukx X

nt k

a^{Œt }.n; k/xⁿC X

nt k t

a^{Œt }.n; k 1/x^nCt Du_kxA^{Œt }_k .x/Cx^tA^{Œt }_{k 1}.x/:

THE GENERALIZED -COMTET NUMBERS 101

Then

A^{Œt }_k .x/Dx^tA^{Œt }_{k 1}.x/

1 u_kx :

Iterating this last recurrence, we obtain (2.1).

From the above relation, we have the following combinatorial expression.

Corollary 1. The generalizedt-Comtet numbers are given by the explicit identity a^{Œt }.n; k/D X

i1Ci2CCikDn t k

uⁱ₁¹uⁱ₂² uⁱ_k^k; (2.2) fornt k.

Theorem 2. The generalizedt-Comtet numbers satisfy the following recurrence relation

a^{Œt }.n; k/D

n t k

X

iD0

uⁱ_ka^{Œt }.n i t; k 1/: (2.3) Proof. Fornt k;

a.n; k/ Du_ka.n 1; k/ Ca.n t; k 1/;

uka.n 1; k/ Du²_ka.n 2; k/ Cuka.n 1 t; k 1/;

u²_ka.n 2; k/ Du³_ka.n 3; k/ Cu²_ka.n 2 t; k 1/;

::: ::: ::: ::: :::

u^{n t k 1}_k a.t kC1; k/Du^{n t k}_k a.t k; k/ Cu^{n t k 1}_k a.t kC1 t; k 1/;

u^{n t k}_k a.t k; k/ Du^{n t k}_k ^C1a.t k 1; k/Cu^{n t k}_k a.t .k 1/; k 1/;

by summing, we get the result.

Theorem 3. We have the following rational explicit formula

a^{Œt }.nCt k; k/D

k

X

jD0

u_j^kCn Q

i¤j.uj ui/; (2.4) which is independent fromt.

Proof. We have

A^{Œt }_k .x/D X

nt k

a^{Œt }.n; k/xⁿDx^{t k}X

n0

a^{Œt }.nCt k; k/xⁿ;

then

X

n0

a^{Œt }.nCt k; k/xⁿD 1

.1 u0x/.1 u1x/ .1 u_kx/

D

k

X

jD0

˛j

1 ujx D

k

X

jD0

u_j^k Q

i¤j.uj ui/ X

n0

u_jⁿxⁿ

DX

n0

0

@

k

X

jD0

u_j^kCn Q

i¤j.uj ui/ 1 Axⁿ;

which gives the result.

Corollary 2. The dual expression depending ont a^{Œt }.n; k/D

k

X

jD0

u_j^{nCk.t 1/}

Q

i¤j.uj ui/: (2.5)

2.1. Exponential generating function for the t-Comtet numbers

Let u1; : : : ; uk be a sequence of complex numbers and let .Am/mD1;:::;n be the sequence of matrices such thatAmismm-matrix

AmD 2 6 6 6 4

u_{k m} u_{k mC1} u_{k 1}

u_{k mC1} u_k

::: : :: : :: 0

uk 1 uk 0 0

3 7 7 7 5

;

with the convention thatu<0D0.

Consider also

j D. 1/^j X

1k₁<k₂<<k_jk

uk₁ uk_j;

(the alternate sequence of elementary symmetric function associated tou1; u2; : : : ; u_k).

We have.p u1/.p u2/ .p u_k/Dp^kC1p^{k 1}C2p^{k 2}C C_k. Now we can state the following lemma which will be used to establish the main result of this subsection.

Lemma 1. We have the following decomposition 1

pⁿ.p u1/ .p u_k/ D

n

X

iD0

˛n i

pⁱ C

k

X

jD1

ˇj

p uj

; (2.6)

THE GENERALIZED -COMTET NUMBERS 103

with˛iD. 1/^b.iC1/=2c

_k^iC1 det.Ai/,˛0D1=_k, andˇj D 1 u_jⁿ

k

Y

iD1 i¤j

.u_j u_i/ .

Proof. We leave the proof to the reader.

LetC_k^{Œt }.x/WD X

nt k

a^{Œt }.n; k/xⁿ

nŠ, withC₀^{Œt }.x/D1. We have

@^t

@x^tC_k^{Œt }.x/D X

nt .k 1/

a^{Œt }.nCt; k/xⁿ nŠ; which gives using relation (1.1),

@^t

@x^tC_k^{Œt }.x/Du_k @^{t 1}

@x^{t 1}C_k^{Œt }.x/CC_{k 1}^{Œt } .x/: (2.7) To solve the linear recurrence differential equation we use Laplace transform. Using the fact that

C_k^{Œt }.0/D @

@xC_k^{Œt }.y/

ˇ ˇ ˇ ˇ_y

D0

D D @^{t 1}

@x^{t 1}C_k^{Œt }.y/

ˇ ˇ ˇ ˇ_y

D0

D0;

we get

k

Y

iD1

.p^t uip^{t 1}/L.C_k^{Œt }.y//DL.C_{k 1}^{Œt } /;

whereL.C_k^{Œt }.y//D Z ₁

0

C_k^{Œt }e^py@y:

Thus by recursion, we get p^{.t 1/k}

k

Y

iD1

.p ui/L.C_k^{Œt }.y//DL.C₀^{Œt }.y//DL.u.y//;

whereu.t /is the Heaviside function. Using Lemma1, we have L.C_k^{Œt }.y///DL.u.y//

2 4

.t 1/k

X

iD0

˛i

pⁱ C

k

X

jD1

ˇj

p uj

3 5: The inverse Laplace transform gives,

C_k^{Œt }.y/D

.t 1/k

X

iD1

˛i

y^{i 1} .i 1/ŠC

k

X

jD1

ˇje^ujy: (2.8)

Theorem 4. The exponential generating function oft-Comtet numbers is given by X

nt k

a^{Œt }.n; k/xⁿ nŠ D

.t 1/k

X

iD1

˛i

x^{i 1} .i 1/ŠC

k

X

jD1

ˇje^ujx: (2.9) 3. THE2-SUCCESSIVE ASSOCIATEDr-WHITNEY NUMBERS

In this section, we study the particular caseukDkmCr. Letn; r0be integers.

Let˘r.n; k/denote the set of partitions of the setŒnCrWD f1; : : : ; n; nC1; : : : ; nC rgintokCrblocks, such that, the firstrelements are in distinct blocks. The elements f1; 2; : : : ; rgwill be called special elements. A block of a partition of the above set is called special if it contains special element. The cardinality of ˘r.n; k/ is the r-Stirling numbers of the second kind [5].

The 2-successive associated r-Whitney numbers of the second kind, denoted Wm;r^Œ2.n; k/, count the number of partitions in˘r.n; k/, such that:

theknon-special blocks contain at least two consecutive numbers,

all the elements but the last one and its predecessor in non-special blocks are coloured with one ofmcolours independently,

the elements in the special blocks are not coloured,

the last elementnCrmust either form a block with its predecessor or belong to another block (special or not-special) satisfying the previous conditions.

We denote by˘_r;m^Œ2.n; k/the set of partitions in˘r.n; k/that satisfying the previous conditions. It is clear that ifrD0andmD1, thenW_1;0^Œ2.n; k/D˚n

k

Œ2, (see [2]).

For example,W_2;3^Œ2.5; 2/D15with the partitions being (themD2different colours of the elements will be fixed asredandblue, and therD3special elements are1; 2 and3):

˚f1g;f2g;f3g;f4; 5; 6g;f7; 8g ; ˚

f1g;f2g;f3g;f4; 5; 6g;f7; 8g ;

˚f1; 6g;f2g;f3g;f4; 5g;f7; 8g ; ˚

f1g;f2; 6g;f3g;f4; 5g;f7; 8g ;

˚f1g;f2g;f3; 6g;f4; 5g;f7; 8g ; ˚

f1; 4g;f2g;f3g;f5; 6g;f7; 8g ;

˚f1g;f2; 4g;f3g;f5; 6g;f7; 8g ; ˚

f1g;f2g;f3; 4g;f5; 6g;f7; 8g ;

˚f1; 8g;f2g;f3g;f4; 5g;f6; 7g ; ˚

f1g;f2; 8g;f3g;f4; 5g;f6; 7g ;

˚f1g;f2g;f3; 8g;f4; 5g;f6; 7g ; ˚

f1g;f2g;f3g;f4; 5; 8g;f6; 7g ;

˚f1g;f2g;f3g;f4; 5; 8g;f6; 7g ; ˚

f1g;f2g;f3g;f4; 5g;f6; 7; 8g ;

˚f1g;f2g;f3g;f4; 5g;f6; 7; 8g : Theorem 5. Forn2k, we have

W_m;r^Œ2.n; k/D.kmCr/W_m;r^Œ2.n 1; k/CW_m;r^Œ2.n 2; k 1/: (3.1)

THE GENERALIZED -COMTET NUMBERS 105

Proof. For any set partition of ˘_r;m^Œ2.n; k/, there are three options: either nCr form a block with its predecessor.nCr 1/, ornCr is in a special block ornCr is in a non-special block. In the first case, there areWm;r^Œ2.n 2; k 1/possibilities.

In the second case, the elementnCr can be place into one of ther special blocks and the remaining elements can be chosen inWm;r^Œ2.n 1; k/. Altogether, we have rWm;r^Œ2.n 1; k/possibilities. For the third case, we can follow a similar argument,

then we obtainkmW_m;r^Œ2.n 1; k/possibilities.

A comparison of (3.1) and (1.1) shows thata^Œ2.n; k/DW_m;r^Œ2.n; k/foru_kDkmC r. Therefore, from Theorem1and Corollary1we get the following corollaries.

Corollary 3. Fork1, W_k^Œ2.x/WD X

n2k

W_m;r^Œ2.n; k/xⁿ

D x^2k

.1 rx/.1 .mCr/x/.1 .2mCr/x/ .1 .kmCr/x/; (3.2) withW₀^Œ2.x/D1 rx¹ . Moreover, the2-successive associatedr-Whitney numbers of the second kind are given by the explicit identity

W_m;r^Œ2.n; k/D X

i0Ci1Ci2CCikDn 2k

rⁱ⁰.mCr/ⁱ¹ .kmCr/^ik; (3.3) forn2k.

In particular, for mD1 and r D0 we obtain the generating function of the 2- successive associated Stirling numbers of the second kind.

Corollary 4. (see [2, Theorem 2.3 and Corollary 2.4] and [3, Theorem 18]) For k1,

Ak.x/WD X

n2k

(n k

)Œ2

xⁿD x^2k

.1 x/.1 2x/ .1 kx/; (3.4) withA0.x/D1. Moreover,

(n k

)Œ2

D X

i1Ci2CCi_kDn 2k

1ⁱ¹2ⁱ² k^ik:

Our next identity expressesWm;r^Œ2.n; k/in terms of˚_i

k

Œ2forin.

Theorem 6. Letn; k0, W_m;r^Œ2.n; k/D

n

X

iD2k

r^{n i} n k n i

! m^{i 2k}

(i k

)Œ2

: (3.5)

Proof. From (3.2) we have X

n2k

W_m;r^Œ2.n; k/xⁿD x^2k

.1 rx/.1 .mCr/x/.1 .2mCr/x/ .1 .kmCr/x/

D x^2k

.1 rx/^kC1 1 _{1 rx}^mx

1 _{1 rx}^2mx

1 _{1 rx}^kmx D.1 rx/^{k 1}

m^2k

mx 1 rx

2k

1 _{1 rx}^mx

1 _{1 rx}^2mx

1 _{1 rx}^kmx D.1 rx/^{k 1}

m^2k

y^2k

.1 y/.1 2y/ .1 kmy/; whereyD_{1 rx}^mx .

Therefore from (3.4), we have X

n2k

W_m;r^Œ2.n; k/xⁿD.1 rx/^{k 1} m^2k

X

i2k

(i k

)Œ2

yⁱ

D X

i2k

(i k

)Œ2

m^{i 2k}xⁱ .1 rx/^{i kC1} D X

i2k

X

j0

m^{i 2k} i kCj i k

! r^j

(i k

)Œ2

x^iCj:

Comparing the coefficients ofxⁿ, we obtain (3.5).

Combinatorial proof:We can construct any set partition of˘r;m^Œ2.n; k/as follows:

we putn i elements in the special blocks. Then there are ^{n k}_{i k}

r^{n i} possibilities.

Note that we have to subtractkelements ofnbecause in the non-special blocks there are at least two consecutive numbers. The remaining i elements (i 2k) can be chosen in m^{i 2k}˚i

k ways. The factor m^{i 2k} accounts for thei 2k non-minimal elements within these blocks that are each to be colored in one ofmways.

From Theorem5and by induction onnwe obtain the following identity.

THE GENERALIZED -COMTET NUMBERS 107

Theorem 7. Forn2kwe have W_m;r^Œ2.n; k/D 1

m^kkŠ

k

X

jD0

. 1/^{k j} k j

!

.mjCr/^{n k}: Proof. Let,

W_m;r^Œ2.n; k/D.mkCr/

m^kkŠ

k

X

jD0

. 1/^{k j} k j

!

.mjCr/^{n 1 k}

C 1 m^{k 1}kŠ

k

X

jD0

. 1/^{k 1 j} k 1 j

!

.mjCr/^{n 1 k}

D.mkCr/

m^{k 1}kŠ

k

X

jD1

. 1/^{k j} k 1 j 1

!

.mjCr/^{n 1 k}

C r m^kkŠ

k

X

jD0

. 1/^{k j} k j

!

.mjCr/^{n 1 k}

D 1 m^kkŠ

n 1 k

X

iD0

r^{n 1 k i}mⁱ 2 4

k

X

jD0

. 1/^{k j}j^iC1 k j

! m

C

k

X

jD0

. 1/^{k j}jⁱr k j

!3 5

D 1 m^kkŠ

k

X

jD0

. 1/^{k j} k j

!

.mjCr/^{n k}:

3.1. Relations with ther-Whitney numbers

Ther-Whitney numbers of the second kindWm;r.n; k/were defined by Mez˝o [17]

as the connecting coefficients between some particular polynomials.

For non-negative integersn; kandrwithnk0and for any integerm > 0 .mxCr/ⁿD

n

X

kD0

m^kWm;r.n; k/x^k; (3.6) wherexⁿDx.x 1/ .x nC1/forn1, andx⁰D1.

Ther-Whitney numbers of the second kind satisfy the recurrence [17]

Wm;r.n; k/DWm;r.n 1; k 1/C.kmCr/Wm;r.n 1; k/: (3.7)

Comparing (3.7) and (3.1) we have the following relation.

Corollary 5. [2, Theorem 4.1] Forn2k,

W_m;r^Œ2.n; k/DWm;r.n k; k/: (3.8) Mez˝o and Ram´ırez [19] studied the r-Whitney matrices of the second and the first kind and they derived several identities for these matrices. In particular, the r-Whitney matrix of the second kind is defined by

ŒWm;r.n; k/_n;k0D 2 6 6 6 6 6 6 6 4

1 0 0 0 0

r 1 0 0 0

r² mC2r 1 0 0

r³ m²C3rmC3r² 3mC3r 1 0

r⁴ m³C4rm²C6r²mC4r³ 7m²C12rmC6r² 6mC4r 1

::: ::: :::

3 7 7 7 7 7 7 7 5 :

Notice that the sequence.W_m;r^Œ2.n; k//_k corresponds with the sequence of elements on rays in direction.1; 1/over ther-Whitney matrix of the second kind.

4. THEt-SUCCESSIVE ASSOCIATEDr-WHITNEY NUMBERS

In this section, we consider the rays in direction.s; 1/, i.e., we are going to study the sequence fWm;r.n sk; k/g. We denote by W_m;r^{Œt } .n; k/the number Wm;r.n sk; k/, wheret DsC1. We call this new sequencethet-successive associatedr- Whitney numbers of the second kind. It is possible to show that the t-successive associatedr-Whitney numbers count the number of partitions in˘r.n; k/, such that:

theknon-special blocks contain at leastt consecutive numbers,

all the elements but the last one and its t 1 predecessors in non-special blocks are coloured with one ofmcolours independently,

the elements in the special blocks are not coloured,

the last elementnCr must either form a block with its t 1-predecessors or belong to another block (special or not-special) satisfying the previous conditions.

Reasoning in a similar manner as in Theorem5we obtain the following results.

Theorem 8. Fornt k, we have

W_m;r^{Œt }.n; k/D.kmCr/W_m;r^{Œt } .n 1; k/CW_m;r^{Œt } .n t; k 1/: (4.1) Fork1,

W_k^{Œt }.x/WDX

n0

W_m;r^{Œt } .n; k/xⁿ

D x^{t k}

.1 rx/.1 .mCr/x/.1 .2mCr/x/ .1 .kmCr/x/; (4.2)

THE GENERALIZED -COMTET NUMBERS 109

withW₀^{Œt }.x/D_{1 rx}¹ . Moreover, fornt kwe have W_m;r^{Œt } .n; k/D 1

m^kkŠ

k

X

jD0

. 1/^{k j} k j

!

.mjCr/^{n .t 1/k}: (4.3) As corollary fortD2we get [3, Theorem 4, Theorem 6 and Theorem 7].

It is not difficult to generalize the relation given in Theorem6.

Theorem 9. Ifn; k0, then W_m;r^{Œt } .n; k/D

n

X

iDt k

r^{n i} n k n i

! m^{i t k}

(i k

)Œt 

: (4.4)

Consequence.From Equation (4.2) we deduce thatWm;r^{Œt }.nC.t 1/k; k/are the classicalr-Whitney numbersWm;r.n; k/.

From the explicit formula given in (4.3) we get the exponential generating function of thet-successive associatedr-Whitney numbers.

Theorem 10. The exponential generating function of thet-successive associated r-Whitney numbers is

W_k^{Œt }.x/WD X

nt k

W_m;r^{Œt } xⁿ nŠ D

k

X

jD0

k j

!. 1/^{k j} kŠm^k

e^{.j mCr/x}

.j mCr/^{.t 1/k}: (4.5) Corollary 6. For the2-successive associatedr-Whitney numbers,

W_k^Œ2.x/D

k

X

jD0

k j

!. 1/^{k j} kŠm^k

e^{.j mCr/x}

.j mCr/^k: (4.6)

These two result are more specified expressions as relation (2.9) of Theorem4.

Proof. (Theorem10) We use the derivation.t 1/ktimes according toxand using the consequence property, we get

@^{.t 1/k}

@^{.t 1/k}xW_k.x/D X

nt k .t 1/k

W_m;r^{Œt } .nCk.t 1/; k/xⁿ nŠ D 1

kŠm^ke^rx.e^mx 1/^k D 1

kŠm^ke^rx

k

X

jD0

k j

!

. 1/^{k j}e^{j mx}

D

k

X

jD0

k j

!. 1/^{k j}

kŠm^k e^{.j mCr/x}:

Theorem 11. Fornk, we have

W_m;r^{Œt } .nC.t 1/k; k/D X

i₁CCi_nDn k i1;;in2f0;1g

n 1

Y

jD0

.rCm.j

j

X

`D1

i_`//^ijC1: (4.7)

Proof. By induction overn, we suppose that the identity is true untiln 1 X

i1CCinDn k i1;;in2f0;1g

n 1

Y

jD0

.rCm.j

j

X

lD1

i_l//^ijC1

D X

i1CCin 1D.n 1/ .k 1/

i1;;in 12f0;1g

n 2

Y

jD0

.rCm.j

j

X

lD1

i_l//^ijC1

C 0 B B

@

X

i1CCin 1D.n 1/ k i1;;in 12f0;1g

n 2

Y

jD0

.rCm.j

j

X

lD1

i_l//^ijC1 1 C C A

.rCmk/:

We have

W_m;r^{Œt } .nC.t 1/k; k/

D.mkCr/W_m;r^{Œt }.n 1C.t 1/k; k/CW_m;r^{Œt } .n 1C.t 1/.k 1/; k/;

which gives the desired result.

Corollary 7. Fornt k, we have W_m;r^{Œt } .n; k/D X

i1CCin .t 1/kDn t k i1;;in .t 1/k2f0;1g

n .t 1/k 1

Y

jD0

.rCm.j

j

X

`D1

i_`//^ijC1; (4.8)

with empty sum equal zero.

Example1. ForkDtD2we have the following formula W_m;r^Œ2.n; 2/D X

i₁CCi_{n 2}Dn 4

rⁱ¹.rCm.1 i1//ⁱ².rCm.2 i1 i2//ⁱ³ .rCm.n 3 i1 i2 in 3//^{in 2};

W_m;r^Œ2.4; 2/D X

iCjD0

rⁱ.rCm.1 i //^j D1;

THE GENERALIZED -COMTET NUMBERS 111

W_m;r^Œ2.5; 2/D X

iCjCkD1

rⁱ.rCm.1 i //^j.rCm.2 i j //^kD3.rCm/;

W_m;r^Œ2.6; 2/D X

iCjCkC`D2

rⁱ.rCm.1 i //^j.rCm.2 i j //^k.rCm.3 i j k//^` D6r²C12mrC7m²:

Theorem 12. We have the following explicit formula

W_m;r^{Œt } .nCt k; k/D 1 m^kkŠ

k j

X

jD0

. 1/^{k j} k j

!

.mjCr/^nCk; (4.9) and thus

W_m;r^{Œt } .n; k/D 1 m^kkŠ

k

X

jD0

. 1/^{k j} k j

!

.mjCr/^{n .t 1/k}: (4.10) Proof. It suffices to setu_kDmkCrthen

uj ui Dm.j i /

in Theorem3.

FortD2we get Theorem7.

Theorem 13. Expression oft-successiver-Whitney numbers in terms of binomials and Stirling numbers.

W_m;r^{Œt }.nCt k; k/Dr m

knCk

X

iD0

nCk i

! r^{n i}mⁱ

(i k

)

: (4.11)

Proof.

W_m;r^{Œt }.nCt k; k/D 1 m^kkŠ

k

X

jD0 nCk

X

iD0

. 1/^{k j} k j

! nCk i

!

.mj /ⁱr^{nCk i}

D 1 m^kkŠ

nCk

X

iD0

mⁱr^{nCk i} nCk i

! _k X

jD0

. 1/^{k j} k j

! jⁱ

D 1 m^k

nCk

X

iD0

mⁱr^{nCk i} nCk i

!(i k

) : Notice thatPk

jD0. 1/^{k j k}_j

jⁱ D0fori < k.

5. Aq-ANALOGUE OF THEt-SUCCESSIVE ASSOCIATEDSTIRLING NUMBERS

Finally, we considerer aq-analogue of thet-successive associated Stirling num- bers of the second kind. For this purpose, we use a similar statistic studied by Carlitz [6], see also [21].

LetDB1=B2= =B_kbe any block representation of a set partition in˘_0;1^{Œt }.n; k/WD

˘^{Œt }.n; k/, with min.B1/ <min.B2/ < <min.B_k/. We define the following stat- istic on the set˘^{Œt }.n; k/.

w^{Œt }./WD

k

X

iD1

.i 1/.jBij tC1/:

We now define theq-analogue of thet-successive associated Stirling numbers of the second kind.

Definition 1. Define˚n k

Œt 

q as the distribution polynomial for thew^{Œt } statistic on the set˘^{Œt }.n; k/, that is,

(n k

)Œt 

q

D X

2˘^{Œt }.n;k/

q^{wŒt ./}; n; k0;

whereqis an indeterminate.

It is clear that˚n k

Œt  1 D˚n

k Œt .

For example, in the set˘^Œ2.7; 3/we have the following partitions:

ff1; 2g;f3; 4; 5g;f6; 7gg; ff1; 2; 3g;f4; 5g;f6; 7gg; ff1; 2; 5g;f3; 4g;f6; 7gg; ff1; 2g;f3; 4g;f5; 6; 7gg; ff1; 2g;f3; 4; 7g;f5; 6gg; ff1; 2; 7g;f3; 4g;f5; 6gg: Therefore,

(7 3

)Œ2

q

Dq⁴Cq³Cq³Cq⁵Cq⁴Cq³D3q³C2q⁴Cq⁵: Let us introduce the following notations.

ŒnqD1CqC Cq^{n 1}; ŒnqŠDŒ1qŒ2q Œnq and

"

n k

#

q

D ŒnqŠ ŒkqŠŒn kqŠ: The last coefficient is calledq-binomial coefficient. IfqD1, thenn

k

1D ⁿ_k . Theorem 14. Fornt k, we have

(n k

)Œt 

q

DŒk_q (n 1

k )Œt 

q

Cq^{k 1} (n t

k 1 )Œt 

q

: (5.1)

Proof. For any set partition of ˘^{Œt }.n; k/, there are two options: either n form a block with its t 1-predecessors, or n is in a block that satisfies the conditions.

In the first case, there are q^{k 1}˚_{n t}

k 1 Œt 

q possibilities. In this case, the size of the last block Bk is t, then this block contributes a factor q^{k 1}. In the second case,

THE GENERALIZED -COMTET NUMBERS 113

the element n can be place into one of the k blocks and thus contributes a factor 1CqCq²C Cq^{k 1}DŒkq. Moreover, the remaining elements can be chosen in

˚n 1 k

Œt 

q ways. Altogether, we haveŒkq˚n 1 k

Œt 

q possibilities.

From above theorem, we obtain the following corollaries.

Corollary 8. Fork1, X

nt k

(n k

)Œt 

q

xⁿD x^{t k}q.^k2/

.1 x/.1 Œ2qx/.1 Œ3qx/ .1 Œkqx/: (5.2) Moreover, theq-analogue of thet-successive associatedr-Stirling numbers are given by the explicit identity

(n k

)Œt 

q

Dq.^k2/ X

i₁Ci₂CCi_kDn t k

Œ1ⁱ_q¹Œ2ⁱ_q² Œkⁱ_q^k; (5.3) fornt k.

Corollary 9. Fornt kwe have (n

k )Œt 

q

D 1 ŒkqŠ

k

X

jD1

. 1/^{k j}

"

k j

#

q

q.^k2j/.Œj q/^{n .t 1/k}:

ACKNOWLEDGEMENT

The authors would like to thank the anonymous referee for some useful comments.

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Authors’ addresses

Hac`ene Belbachir

USTHB, Faculty of Mathematics, RECITS Laboratory. P.B. 32, El Alia, 16111, Bab Ezzouar, AL- GERIA

E-mail address:hbelbachir@usthb.dz; hacenebelbachir@gmail.com

Carolina Forero

Universidad Sergio Arboleda, Departamento de Matem´aticas, Bogot´a, COLOMBIA E-mail address:tcarolinafc@gmail.com

Jos´e L. Ram´ırez

Universidad Nacional de Colombia, Departamento de Matem´aticas, Bogot´a, COLOMBIA E-mail address:jlramirezr@unal.edu.co