**2.1 Fundament for 2**�**summands**

Let *F* be a field [4], and *F X*½ � the ring of polynomials in the indeterminate x, by the generating set of

$$F[X] = \mathcal{K}\left(\left\{\bigcup\_{j \in \mathbb{N}\_0} \varkappa^j\right\} \bigcup \{1\}\right) \tag{1}$$

Recall the mapping from [2]

*DOI: http://dx.doi.org/10.5772/intechopen.91422*

by the equation

the half-open interval,

a function

**101**

By [7] we have

the following applications:

Let *x*∈<sup>þ</sup> and *J* be an index set; define

<sup>¼</sup> <sup>X</sup> *n*^ 1

8 < :

*δn*^¼1

⋯ X

*<sup>n</sup>*^�^ ð Þ*<sup>k</sup>* <sup>þ</sup>1k*δ*^

*δ*^ *<sup>k</sup>*¼1

� � <sup>X</sup>

*ϕ <sup>n</sup>*^�^ ð Þ*<sup>k</sup>* <sup>þ</sup>1k*δϕ*þ<sup>1</sup>

Under this outline, we would have the following ordering:

*i*< LEX *j*< LEX *k* � �< LEX *z*< LEX *zh*< LEX *zi* � �< LEX *z*< LEX *zzh*< LEX *zzi*< LEX

them in order to be used in an algorithm.

**Lemma 2.3.** *Existence of a choice function.*

such that *c B*ð Þ is an element of *B*, for each *B* ∈B.

residue classes which have a multiplicative inverse in *=n*:

*δϕ*¼1

where the widehat script in the binomial terms (*n*^, ^

*n*^ ^ *k* � �

where

*τ* : <sup>0</sup> � <sup>0</sup> ! ⋃

^ *k*

*k*þ1

⋯ X 2 *<sup>n</sup>*^�^ ð Þ*<sup>k</sup>* <sup>þ</sup>1k*<sup>l</sup>*

*k*¼1

� �*i* ¼

*<sup>∂</sup>*P*n*ð Þ *<sup>F</sup>* <sup>∈</sup>*π*<sup>þ</sup>

*Alternative Representation for Binomials and Multinomies and Coefficient Calculation*

**<sup>A</sup>** *<sup>j</sup>* <sup>⊂</sup><sup>þ</sup> <sup>∍</sup> *<sup>n</sup>*

X 1 *<sup>n</sup>*^�^ ð Þ*<sup>k</sup>* <sup>þ</sup>1k*<sup>k</sup>*

*j*¼1

X *ϕ <sup>n</sup>*^�^ ð Þ*<sup>k</sup>* <sup>þ</sup><sup>1</sup>

8 >>>>>>><

>>>>>>>:

those from the index set of the summation sequences, and provided that *δϕ* represents the set of a sequence of consecutive characters in the lexicographic order, by

It is of great importance that we could establish a counting relation between

*c* : B ! ⋃

Given a collection [5] B of nonempty sets (not necessarily disjoint), there exists

A bijection between *δϕ* and <sup>þ</sup> can now be established. Consider the collection of

when *m* and *n* are relative prime integers. However, we will set *n* ¼ 19, to define

by *g x*ð Þ ≔ mod ð Þ! *x*, 19 f g *i*, *j*, *k*, … , *z*

*B*∈B

ð Þ *=n* � <sup>¼</sup> *<sup>a</sup>*∈*=n* : <sup>∃</sup>*c*∈*=n*∍*<sup>a</sup>* � *<sup>b</sup>* <sup>¼</sup> <sup>1</sup> � �⊂*=n* (12)

ð Þ *=mn* � ffi ð Þ *=m* � � ð Þ *=n* � (13)

*g* : ð Þ *=*19 � ! f g *i*, *j*, *k*, … , *z* (14)

*δϕ*¼1

X *ϕ δϕ*þ<sup>1</sup>

*δϕ*¼1

*δϕ* ≔ f g ½ Þ *i*, *j*, *k*, … *z*, *zh*, *zi*, … , þ∞ : 1≤*ϕ* < þ ∞; *ϕ*∈ <sup>þ</sup> ⊂<sup>þ</sup> (9)

*k*

X 0 1

*i*¼1 *ii* " #

> *<sup>ϕ</sup>* <sup>¼</sup> ^ *k*

*<sup>ϕ</sup>* 6¼ ^ *k*

� �↦*<sup>τ</sup>* (6)

9 = ;

(7)

(8)

*j*,*k*,*::*, *δ*^ *<sup>k</sup>* ,…, *δn*^

*k*) was placed to distinguish

⋯ (10)

*B* (11)

*j*¼1

Now, denote the set of polynomials with positive coefficients of degree *∂* at most *n* over *F*, by

$$\pi\_+ := \left\{ \mathcal{P}\_n(F) \subseteq F[X] \subset \mathbb{R}^{\circ \circ} : \mathcal{P}\_n(F) = \left( \sum\_{1 \le \phi \in \mathbb{Z}\_+} \mathbb{1}\_{\{\phi\}} \right)^n, \mathcal{P}\_j(F) \bigcap \mathcal{P}\_{j+1}(F) = \mathcal{Q} \right\} \tag{2}$$

and let **A** be the set of the coefficients of those polynomials:

$$\omega\_{\omega} := \bigcup\_{\forall k, n \in \mathbb{N}} \pi\_+ \backslash \prod\_{j \in \{1, 2\}} \mathfrak{x}\_j^{(2-j)n + (2j-3)k} \tag{3}$$

**Definition 2.1.** A subset *A* of the real numbers is said to be **inductive** if it contains the number 1, and for every *x* in *A*, the number *x* þ 1 is also in *A*. Let C be the collection of all inductive subsets of . Then the set [5] <sup>þ</sup> of **positive integers** is defined by the equation

$$\mathbb{Z}\_{+} \coloneqq \bigcap\_{A \in \mathcal{C}} A \subset \mathbb{R}\_{+}. \tag{4}$$

**Definition 2.2. Lexicographic ordering:** > LEX � �*.* We say [6] *p*> LEX *q*, if we have the following conditions:

$$\begin{aligned} &p > q \Leftrightarrow \exists i \in \mathbb{Z}\_{+} \,\,\,\,\Rightarrow \,a\_{i} \neq b\_{i} \wedge a\_{\min\{i\}} > b\_{\min\{i\}}\\ &\Rightarrow \bigcup\_{n \in \mathbb{Z}\_{+}} \{\{a\_{n}\}\backslash\{b\_{n}\}\} > 0 \end{aligned} \tag{5}$$

for some non-zero entry, i.e., for some *n*∈þ.

*Alternative Representation for Binomials and Multinomies and Coefficient Calculation DOI: http://dx.doi.org/10.5772/intechopen.91422*

Recall the mapping from [2]

$$\pi: \mathbb{N}\_0 \times \mathbb{N}\_0 \to \bigcup\_{j=1}^{\partial \mathcal{P}\_n(F) \in \pi\_+} \mathbb{N}\_j \subset \mathbb{Z}\_+ \twoheadrightarrow \binom{n}{k} \mapsto \pi \tag{6}$$

by the equation

$$\begin{pmatrix} \hat{\boldsymbol{n}} \\ \hat{\boldsymbol{k}} \end{pmatrix} = \left\{ \sum\_{\delta\_{\hat{n}}=1}^{1} \cdots \sum\_{\delta\_{\hat{k}}=1}^{\left(\hat{n}-\delta\right)+1\|\delta\_{\hat{k}+1}} \cdots \sum\_{\delta\_{\hat{k}}=1}^{\left(\hat{n}-\delta\right)+1\|\boldsymbol{l}} \sum\_{j=1}^{\left(\hat{n}-\delta\right)+1\|\boldsymbol{k}} \left[ \sum\_{i=1}^{1} \boldsymbol{i}\_{i} \right]\_{j,\boldsymbol{k},...,\delta\_{\hat{k}},...,\delta\_{\hat{k}}} \right\} \tag{7}$$

where

This document comprises the extension of the results obtained from [2]. This extension will based upon formula (4): It will be explained in more detail, oriented towards the developing of an algorithm to perform the calculations; this formula will be extended to obtain a general one to achieve the calculation of multinomial coefficients, using the procedure from [2]; a proof for this general formula will be given. The theoretical application of formula (4) will be shown in some other case study, taking over a lemma from [3] and developing an alternative proof by these means. As it was mentioned above, two algorithms based on this result are worked up and were implemented as a program in a computer algebra system (CAS). For its best understanding, this chapter was in general written in the order just mentioned above. For the reader's convenience, and in order to get a broad understanding of this chapter, it is highly advisable to give a read at the original

*Mathematical Theorems - Boundary Value Problems and Approximations*

Let *F* be a field [4], and *F X*½ � the ring of polynomials in the indeterminate x,

*j* ∈ <sup>0</sup> *xj* ( )

Now, denote the set of polynomials with positive coefficients of degree *∂* at

1≤ *ϕ*∈ <sup>þ</sup> ≤ *s*

*<sup>π</sup>*þn <sup>Y</sup> *j*∈f g 1, 2

**Definition 2.1.** A subset *A* of the real numbers is said to be **inductive** if it contains the number 1, and for every *x* in *A*, the number *x* þ 1 is also in *A*. Let C be the collection of all inductive subsets of . Then the set [5] <sup>þ</sup> of **positive integers**

> <sup>þ</sup> ≔ ⋂ *A* ∈C

*q*, if we have the following conditions:

( )

!*<sup>n</sup>*

!

*x*f g*<sup>ϕ</sup>*

*x*

LEX � � *.*

f g f gn *an* f g *bn* <sup>&</sup>gt;<sup>0</sup> (5)

*q* ⇔ ∃*i* ∈<sup>þ</sup> ∍ *ai* 6¼ *bi* ∧ *a*min f g*<sup>i</sup>* >*b*min f g*<sup>i</sup>*

ð Þ 2�*j n*þð Þ 2*j*�3 *k*

,P*j*ð Þ *F* ⋂ P *<sup>j</sup>*þ1ð Þ¼ *F* ∅

*<sup>j</sup>* (3)

*A* ⊂ þ*:* (4)

⋃ f g1

(1)

(2)

*F X*½ �¼ L ⋃

and let **A** be the set of the coefficients of those polynomials:

**A** ≔ ⋃ ∀*k*, *n*∈

**Definition 2.2. Lexicographic ordering:** >

*p*> LEX

> ) ⋃ *n*∈<sup>þ</sup>

for some non-zero entry, i.e., for some *n*∈þ.

document [2].

**2. Mathematical framework**

**2.1 Fundament for 2**�**summands**

*<sup>π</sup>*<sup>þ</sup> <sup>≔</sup> <sup>P</sup>*n*ð Þ *<sup>F</sup>* <sup>⊆</sup>*F X*½ �<sup>⊂</sup> <sup>∞</sup> : <sup>P</sup>*n*ð Þ¼ *<sup>F</sup>* <sup>X</sup>

by the generating set of

most *n* over *F*, by

is defined by the equation

We say [6] *p*>

**100**

LEX

$$\cdots \cdot \sum\_{\delta\_{\phi}=1}^{\left(\hat{n}-\hat{k}\right)+1\left\Vert\delta\_{\phi+1}\right\Vert} \cdot \dot{\imath} = \begin{cases} \binom{\hat{n}-\hat{k}}{\sum}^{+1} & \phi = \hat{k} \\\\ \sum\_{\delta\_{\phi}=1}^{\delta\_{\phi}} & \phi \neq \hat{k} \end{cases} \tag{8}$$

where the widehat script in the binomial terms (*n*^, ^ *k*) was placed to distinguish those from the index set of the summation sequences, and provided that *δϕ* represents the set of a sequence of consecutive characters in the lexicographic order, by the half-open interval,

$$\delta\_{\phi} := \{ [i, j, k, \dots z, zh, zi, \dots, +\infty) : 1 \le \phi < +\infty; \phi \in \mathbb{Z}\_+ \} \subset \mathbb{Z}\_+ \tag{9}$$

Under this outline, we would have the following ordering:

$$\mathbf{i} < j < k \cdot\_{\text{LEX}} < z < z \\ \mathbf{h} < z \mathbf{i} < \dots < z < z \\ \mathbf{z} \mathbf{h} < z \\ \mathbf{z} \mathbf{i} < \dots \\ \mathbf{0} \tag{10}$$

It is of great importance that we could establish a counting relation between them in order to be used in an algorithm.

#### **Lemma 2.3.** *Existence of a choice function.*

Given a collection [5] B of nonempty sets (not necessarily disjoint), there exists a function

$$\mathcal{L}: \mathcal{J} \mathcal{B} \to \bigcup\_{\mathcal{B} \in \mathcal{A}} \mathcal{B} \tag{11}$$

such that *c B*ð Þ is an element of *B*, for each *B* ∈B.

A bijection between *δϕ* and <sup>þ</sup> can now be established. Consider the collection of residue classes which have a multiplicative inverse in *=n*:

$$(\mathbb{Z}/n\mathbb{Z})^{\times} = \left\{ \overline{a} \in \mathbb{Z}/n\mathbb{Z} : \exists \overline{c} \in \mathbb{Z}/n\mathbb{Z} \exists \overline{a} \cdot \overline{b} = \overline{1} \right\} \subset \mathbb{Z}/n\mathbb{Z} \tag{12}$$

By [7] we have

$$\left(\mathbb{Z}/m\mathbb{Z}\mathbb{Z}\right)^{\times} \cong \left(\mathbb{Z}/m\mathbb{Z}\right)^{\times} \times \left(\mathbb{Z}/n\mathbb{Z}\right)^{\times}\tag{13}$$

when *m* and *n* are relative prime integers. However, we will set *n* ¼ 19, to define the following applications:

Let *x*∈<sup>þ</sup> and *J* be an index set; define

$$\begin{aligned} \mathbf{g} : \left( \mathbb{Z}/\mathbf{19} \mathbb{Z} \right)^{\times} &\to \{ i, j, k, \dots, z \} \\ \mathbf{b} \circ \mathbf{g}(\mathbf{x}) &\coloneqq \text{mod } (\mathbf{x}, \mathbf{19}) \to \{ i, j, k, \dots, z \} \end{aligned} \tag{14}$$

By the Lemma 2.3 we can construct

$$f: \mathbb{Z}\_{+} \to \bigcup\_{j \in J} \{\mathbf{z}\}\_{j}$$

$$\mathbf{b} \mathbf{y} f(\mathbf{x}) \coloneqq \bigcup\_{j \in \mathcal{K}\left(\big|\mathbf{z}\big|\right)} \{\mathbf{z}\big|\big)\_{j}\tag{15}$$

**Corollary 2.6.** The set <sup>þ</sup> � <sup>þ</sup> is countably infinite [8].

infinite.

*δ* ∗ *<sup>ϕ</sup>* ≔ *n*^, *i*

where *f*

unique function

X*δ* ∗ 0 *δ* ∗ <sup>1</sup> ¼0

**103**

X *δ* ∗ 0 1

8 >< >:

⋯ X

*δ* ∗ <sup>0</sup> �*<sup>δ</sup>* <sup>∗</sup> ð Þ <sup>1</sup> <sup>þ</sup>1k*δδ* <sup>∗</sup>

> *δ* ∗ 1

*δδ* <sup>∗</sup> <sup>1</sup> <sup>¼</sup><sup>1</sup> <sup>1</sup> <sup>þ</sup><sup>1</sup>

*δδ* <sup>∗</sup> <sup>0</sup> <sup>¼</sup><sup>1</sup>

such that *h*ð Þ¼ 1 *a*0,

**2.2 Extension to** *n*�**summands**

defined in similar way as *δϕ*:

<sup>∗</sup> , *j*

superscript on the alphabetic letters.

lection of summands; set the following:

*<sup>φ</sup>*f g *<sup>s</sup>*�*<sup>f</sup>* <sup>¼</sup> <sup>X</sup> *δ* ∗

*<sup>f</sup>* <sup>¼</sup> <sup>0</sup> *<sup>λ</sup>*¼<sup>0</sup> <sup>≤</sup>*ϕλ* <sup>≤</sup>*<sup>δ</sup>* <sup>∗</sup>

0≤*λ*∈ <sup>0</sup> ≤1∀*λ*

� � �

respectively. Continue recursively performing the expansion:

⋯ X 2

*k*¼1

*δ* ∗ <sup>0</sup> �*<sup>δ</sup>* <sup>∗</sup> ð Þ <sup>1</sup> <sup>þ</sup>1k*<sup>l</sup>*

*f*�1

� � � *λ*¼1

Subindices f g*f* , f g *s* � *f* ∈<sup>þ</sup> represent a consecutive number of a summand and the amount of remaining summands after the binomial theorem expansion,

extended; that is why we will do it here.

*DOI: http://dx.doi.org/10.5772/intechopen.91422*

Finally, in this corollary, it follows that the tuples <sup>þ</sup> � *δϕ* ⊂ <sup>þ</sup>

*Alternative Representation for Binomials and Multinomies and Coefficient Calculation*

Now that the countability of the collection *δϕ* was explained and a bijective function settled down to perform it, we will proceed to extend (1) to *n*� summands. Note that in [2] this formula remained unaltered in this sense, so that it was not

For the foremost part, another index set on a half-open interval is introduced,

The need of another index set comes from the fact that in the extension to the

◦ *<sup>g</sup>*<sup>0</sup> � � : <sup>þ</sup> ! *<sup>δ</sup>* <sup>∗</sup>

**Theorem 2.7. Principle of recursive definition**. Let *A* be a set; let *a*<sup>0</sup> be an element of *A*. Suppose *ρ* is a function that assigns to each function *f* mapping a nonempty section of positive integers into *A*, an element of *A*. Then there exists a

Now the same method developed in [2] will be performed, following the binomial theorem [10–12] and the recursive principle: Let *φ*, *γ* ∈P*n*ð Þn *F* **A** be two col-

> *ϕ*j *λ*¼1 *ϕ*j *λ*¼0 � �*<sup>γ</sup>*

> > X 1

*j*¼1

*δ* ∗ <sup>0</sup> �*<sup>δ</sup>* <sup>∗</sup> ð Þ <sup>1</sup> <sup>þ</sup>1k*<sup>k</sup>*

<sup>0</sup> and *g*<sup>0</sup> are defined the same way as in the former, regarding ∗ as a

*n*� summands approach, a second lawyer of summands sequences and a new sequence of multipliers will arise; this way index scrambling between the two lawyers will be avoided. Then in a similar fashion, the modified function will apply:

> *f* 0

<sup>∗</sup> , *<sup>k</sup>* <sup>∗</sup> , … *<sup>z</sup>* <sup>∗</sup> , *zh*<sup>∗</sup> , *zi* <sup>∗</sup> f g ½ Þ , … , <sup>þ</sup><sup>∞</sup> : <sup>0</sup>≤*ϕ*<sup>&</sup>lt; <sup>þ</sup> <sup>∞</sup>; *<sup>ϕ</sup>*∈<sup>þ</sup> <sup>⊂</sup><sup>þ</sup> (19)

� � are countably

*<sup>ϕ</sup>* (20)

*h* : <sup>þ</sup> ! *A* (21)

*h i*ðÞ¼ *ρ*ð Þ *h*jf g 1, … , *i* � 1 ∀*i*>1*:* (22)

*<sup>ϕ</sup>*j j *<sup>λ</sup>*¼1�*<sup>ϕ</sup> <sup>λ</sup>*¼<sup>0</sup> *<sup>f</sup> <sup>φ</sup><sup>ϕ</sup>*<sup>j</sup>

X 0 1

*i*¼1 *ii* " #

*j*,*k*,*::*, *δδ* <sup>∗</sup>

<sup>1</sup> , … , *δδ* <sup>∗</sup> 0

*λ*¼0

f g *<sup>s</sup>*�*<sup>f</sup>* (23)

9 >= >; *γ δ* ∗ <sup>0</sup> �*<sup>δ</sup>* <sup>∗</sup> 1 <sup>1</sup> *φδ* <sup>∗</sup> 1 *s*�1

(24)

Then the composition *<sup>f</sup>* ◦ *<sup>g</sup>* will be the searched function:

$$(f \circ \mathbf{g}) : \mathbb{Z}\_{+} \to \delta\_{\phi}$$

$$\text{by } (f \circ \mathbf{g})(\mathbf{x}) = \bigcup\_{j \in \mathcal{K}\left(\lfloor \frac{x}{\mathfrak{M}} \rfloor\right)} \{\mathbf{z}\}\_{j} \bigcup \{Im(\mathbf{g})\} \tag{16}$$

This could be graphically represented in **Figure 1**.

**Definition 2.4.** A set *A* is said to be infinite if it is not finite [8]. It is said to be countably infinite if there is a bijective correspondence:

$$f: \mathcal{A} \to \mathbb{Z}\_+\tag{17}$$

In the next theorem, the collection *δϕ* is countable:

**Theorem 2.5.** Let f g *En* , *n* ¼ 1, 2, 3, …, be a sequence of countable sets, and put

$$S = \bigcup\_{n=1}^{\infty} E\_n \tag{18}$$

Then S is countable [9].

Since it was possible to establish a bijection between <sup>þ</sup> and the collection *δϕ*, and since each summation in (1) corresponds to just one element of *δϕ*, it follows that the summands in (1) are also countable.

**Figure 1.** *Graphical representation of the mapping f* ð Þ ◦ *<sup>g</sup>* : <sup>þ</sup> ! *δϕ*.

*Alternative Representation for Binomials and Multinomies and Coefficient Calculation DOI: http://dx.doi.org/10.5772/intechopen.91422*

**Corollary 2.6.** The set <sup>þ</sup> � <sup>þ</sup> is countably infinite [8].

Finally, in this corollary, it follows that the tuples <sup>þ</sup> � *δϕ* ⊂ <sup>þ</sup> � � are countably infinite.
