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arXiv:0908.3248v1 [math.CO] 22 Aug 2009

Generalization of Fibonomial Coefficients

M. Dziemia´ nczuk1

Abstract

Following Lucas and then other Fibonacci people Kwa´ sniewski had introduced and had

started ten years ago the open investigation of the overall F-nomial coefficients which en-

compass among others Binomial, Gaussian and Fibonomial coefficients with a new unified

combinatorial interpretation expressed in terms of cobweb posets’ partitions and tilings of

discrete hyperboxes. In this paper, we deal with special subfamily of T -nomial coefficients.

The main aim of this note is to develop the theory of T -nomial coefficients with the help

of generating functions. The binomial-like theorem for T -nomials is delivered here and some

consequences of it are drawn. A new combinatorial interpretation of T -nomial coefficients is

provided and compared with the Konvalina way of objects’ selections from weighted boxes.

A brief summary of already known properties of F-nomial coefficients is served.

This is The Internet Gian-Carlo Rota Polish Seminar article, No 9, Subject 5, 2009-08-08,

http://ii.uwb.edu.pl/akk/sem/sem rota.htm

1 Introduction

At first let us recall definition of the F-nomial coefficients and summarize already known prop-

erties of these arrays of nonnegative integer numbers. Next, a special family of tileable sequences

T and its corresponding T -nomial coefficients is being considered.

Definition 1.1. Let F stays for a natural numbers’ sequence {nF}n≥0, and n,k ∈ N, such that

n ≥ k. Then F-nomial coefficient is identified with the symbol

?n

F

where nF! = nF(n − 1)F···1F and nk

Since F ≡ {nF}n≥1stays for a sequence of natural numbers, i.e. nF = n, the F-nomial

coefficient reduce to ordinary binomial coefficient

k

?

F= nF(n − 1)F···(n − k + 1)F with 0F! = n0

=

nF!

kF!(n − k)F!=nk

F

kF!

(1.1)

F= 1.

?n

k

?

F

=

n!

k!(n − k)!=

?n

k

?

For a sequence F of next Fibonacci numbers {Fn}n≥0we obtain Fibonomial coefficient, i.e.

?n

F

Finally, if an n-th element of the sequence F is nF= nq= (qn−1)/(q−1) we obtain q-binomial

(Gaussian) coefficient

?n

F

k

?

=

Fn!

Fk!Fn−k!=

?n

k

?

Fib

k

?

=

nq!

kq!(n − k)q!=

?n

k

?

q

.

1Institute of Informatics, University of Gda´ nsk, Poland; E-mail: mdziemianczuk@gmail.com.

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Let us review main properties of F-nomial coefficients, where F denote any natural numbers’

valued sequence. According to the definition, we have

1. Complementation Rule

?n

k

?

F

=

?

n

n − k

?

F

, (1.2)

2. Iterative Rule

?n

m

?

F

?m

k

?

F

=

?n

k

?

F

?n − k

n − m

?

F

(1.3)

3. Multinomial coefficients

?

n

i1,i2,...,ik

?

F

=

?n

i1

?

F

?n − i1

i2

?

F

···

?n − i1− ··· − ik−1

ik

?

F

(1.4)

4. Inversion formula

?n

k

?−1

F

=

?n

k

?

F

n−k

?

s=1

(−1)s

?

i1+i2+···+is=n−k

i1,i2,...,is≥1

?

n − k

i1,i2,...,is

?

F

(1.5)

while?n

Since a special A-admissible family of natural numbers’ valued sequences F, introduced by

Kwa´ sniewski [8, 6, 7] is taken into account, the F-nomial coefficients counts blocks of cobweb

poset’s partitions. This family includes for example Natural numbers, Gaussian and Fibonacci

integers.

n

?−1

F

= 1.

Combinatorial Interpretation I (“Partitions of cobweb layer”)

Let ?Φk→Φn? be a cobweb poset layer with m levels Φ, where k = n − m + 1, and Cmax(Π)

denote the number of maximal chains of a poset Π. Suppose that F is an admissible sequence

A. Then

?Φn→Φk? =

i=1

λ?

πi

⇔

λ =

?n

m

?

F

(1.6)

while Cmax(πi) = Cmax(?Φ1→Φm?) and πi∩ πj= ∅ for any i ?= j.

Kwa´ sniewski posed also the cobweb tiling problem [6], where one asks about family of so-

called tileable sequences T . For such sequences, the F-nomial coefficients obtain additional

combinatorial interpretation with respect to the general Interpretation I.

Combinatorial Interpretation II (“Tilings of hyper F-boxes”)

Let Vk,ndenote an m-dimensional discrete box A1×A2×···×Amwhere |As| = (k+s−1)F

and m = n − k + 1 (see: [11, 14, 13]). Suppose F is a tileable sequence T . Then the value of

T -nomial coefficient?n

For further reading about combinatorial interpretations we refer the reader to [6, 8, 13, 14, 15]

and to the references therein.

m

?

Tis equal to the number of m-dimensional translates bricks V1,mthat

form a tiling of Vk,n.

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2 Tileable sequences and T -nomial coefficients

The Kwa´ sniewski upside-down notation [6, 7], invented in the spirit of Knuth [2] is being

here taken for granted. For example n-th element of a sequence F = {nF}n≥1 is Fn ≡ nF,

consequently nF! = nF·(n−1)F···1F, and nk

In this section we define so-called tileable sequences Tp,qfor arbitrary p and q. In the next

part of this paper, the corresponding T -nomial coefficients, determined by these sequences are

considered.

F= nF·(n−1)F···(n−k+1)Fwith n0

F= 0F! = 1.

Definition 2.1. A natural numbers’ valued sequence T ≡ {nT}n≥1constituted by n-th coeffi-

cient of the generating function Tp,q(x) expansion, i.e., nT = [xn]Tp,q(x), where

Tp,q(x) = 1T·

x

(1 − px)(1 − q x)

(2.1)

while p,q ∈ R and 1T ∈ R is called tileable and denoted by Tp,q.

Take a tileable sequence T ≡ Tp,q. Without loss of generality we assume that 1T = 1. Let

n ∈ N be given. Then for any m,k ∈ N such that m + k = n an n-th element of T satisfies the

following recurrence relation

nT = (k + m)T = pmkT+ qkmT. (2.2)

We may generalize the above as follows. Let b be a composition ?b1,b2,...,bk? of the number

n into k non-zero parts. Then an n-th element of the sequence T satisfies

k

?

An explicit formula for n-th term of T is given by

for n ≥ 1. In the other hand we have

n

?

The explicit formula of T -nomial coefficients while the sequence T = Tp,qis

Due to (2.2) we can show that T -nomial coefficients satisfy binomial-like recurrence relation,

i.e., for any n,k ∈ N we have

?n

T

with initial values?n

3

nT =?b1+ b2+ ··· + bk

?

T=

i=1

p(bi+1+···+bk)q(b1+···+bi−1)(bi)T. (2.3)

nT =

qn−pn

q−p,q ?= p,

nqn−1,q = p.

(2.4)

nT =

i=1

q(n−i)p(i−1). (2.5)

?n

k

?

T

=

?k

?n

i=1

(pn−i+1−qn−i+1)

(pi−qi)

,p ?= q,

k

?pk(n−k),q = p.

(2.6)

k

?

= pn−k

?n − 1

k − 1

?

T

+ qk

?n − 1

k

?

T

(2.7)

0

?

T=?n

n

?

T= 1.

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3 Generating functions of T -nomial coefficients

In the sequel, T stand for a tileable sequence Tp,q.

Theorem 1. Let An(x) and Bn(x) be ordinary generating functions

An(x) =

?

k≥0

(−1)kq(k

2)p(k

2)

?n

k

?

?

T

xk, (3.1a)

Bn(x) =

?

k≥0

?n + k − 1

k

T

xk. (3.1b)

Then An(x) and Bn(x) are given as follows

An(x) =

n

?

n

?

i=1

?

1 − q(i−1)p(n−i)x

?

(3.2a)

Bn(x) =

i=1

1

?1 − q(i−1)p(n−i)x?

(3.2b)

with A0(x) = 1 and B0(x) = 0.

Proof. It is a simple exercise using ordinary generating functions. We apply recurrence (2.7) to

(3.1a) and (3.1b) to see that

An(x) =?1 − p(n−1)x?An−1(q x),

Bn(x) =

A0(x) = 1;

1

(1−p(n−1)x)Bn−1(q x),

B0(x) = 0,B1(x) =

1

1−x.

And then it follows immediately that (3.2a) and (3.2b) hold by induction.

?

We can infer also another form of the generating function An(x) of T -nomial coefficients.

Let Cn(x) be a generating function defined as follows

Cn(x) =

?

k≥0

(−1)kq(k

2)p(n−k

2)

?n

k

?

T

. (3.3)

Then simple calculation using recurrence (2.7) yields

Cn(x) = pn−1(1 − x)Cn−1(xp/q)

which immediately results in

Cn(x) =

n

?

i=1

?

p(i−1)− q(i−1)x

?

(3.4)

for n ≥ 1 and C0(x) = 1.

Corollary 1. Let k,n ∈ N be given. Then the value of T -nomial coefficient is equal

?n + k − 1

T

k

?

=

?

1≤b1≤···≤bk≤n

λn

b1λn

b2···λn

bk, (3.5)

where λn

i= q(i−1)p(n−i)for i = 1,2,...,n.

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Proof. Let us consider generating function (3.2b). From Theorem 1 we have

?n + k − 1

k

?

T

= [xk]

n

?

?

i=1

1

?1 − q(i−1)p(n−i)x? = [xk]

cn,kxk= cn,k

n

?

i=1

?

j≥0

q(i−1)jp(n−i)jxj

= [xk]

k≥0

where coefficients cn,ktake a form

cn,k=

?

a1+a2+···+an=k

a1,a2,...,an≥0

?

a1,a2,...,an≥0

q(1−1)a1p(n−1)a1q(2−1)a2p(n−2)a2···q(n−1)anp(n−n)an

=

a1+a2+···+an=k

?

q0p(n−1)?a1?

q1p(n−2)?a2···

?

q(n−1)p0?an.

Notice that there are at most k non-zero variables aiin the sum a1+ a2+ ··· + an= k, where

ai ≥ 0, for i = 1,2,...,n. Therefore, for fixed k we choose a multiset B (with repetition

allowed) of k these variables aithat are non-zero, i.e. B = {b1,b2,...,bk} where 1 ≤ bi≤ n and

1 ≤ b1≤ b2≤ ··· ≤ bk≤ n. Hence

cn,k=

?

?

1≤b1≤b2≤···≤bk≤n

?

λn

q(b1−1)p(n−b1)??

b1λn

q(b2−1)p(n−b2)?

···

?

q(bk−1)p(n−bk)?

=

1≤b1≤b2≤···≤bk≤n

b2···λn

bk

(3.6)

where λn

i= q(i−1)p(n−i). Hence the thesis.

?

Corollary 2. Let k,n be natural numbers. Then the following hold

?n

k

?

T

q(k

2)p(k

2)=

?

1≤b1<b2<···<bk≤n

λn

b1λn

b2···λn

bk

(3.7)

with λn

i= q(i−1)p(n−i)for i = 1,2,...,n.

Proof. Consider generating function (3.2a). It is easy to see that

n

?

i=1

(1 − λn

ix) =

n

?

k≥0

?

1≤b1<b2<···<bk≤n

λn

b1λn

b2···λn

bk

(−1)kxk

where bidenote indices of chosen factors (1 − λn

proof.

ix). Combining (3.1a) with (3.2a) finishes the

?

Since p and q are natural numbers, Corollary 1 and Corollary 2 provide a combinatorial

interpretation expressed in the language of object selections’ from weighted boxes, where weight

of k-th of n boxes is given by λn

k. Let us sum up the above in the following corollary. Compare

it also with the Konvalina [3, 4] unified interpretation of Binomial, Gaussian coefficients and

Stirling numbers.

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