The Twelve Triangular Matrix Forms of the Pascal Triangle: a Systematic Approach with the Set of Circulant Operators B

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The Twelve Triangular Matrix Forms of the Pascal Triangle: a Systematic Approach with the Set of Circulant Operators B. Birregah, Prosper Doh, Kondo Hloindo Adjallah

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B. Birregah, Prosper Doh, Kondo Hloindo Adjallah. The Twelve Triangular Matrix Forms of the Pascal Triangle: a Systematic Approach with the Set of Circulant Operators. 10th WSEAS International Confenrence on Applied Mathematics, WSEAS, Nov 2006, Dallas, Texas, United States. pp.220-225. hal-03091097

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The Twelve Triangular Matrix Forms of the Pascal Triangle: a

Systematic Approach with the Set of CirculantKONDO Operators H. ADJALLAH PROSPER K. DOH University of Technology of Troyes BABIGA BIRREGAH University of Nancy 2 Institute Charles Delaunay University of Lome´ 23, boulevard Albert 1er - BP 3397 12 rue Marie Curie, College of Science, F54015 Nancy Cedex F.10010 Troyes Cedex Applied Math. Dpt. BP. 1515 FRANCE FRANCE TOGO

Abstract: This work is devoted to a systematic investigation of triangular matrix forms of the Pascal Triangle. To start, the twelve matrix forms (collectively referred to as G-matrices) are presented. To highlight one way in which the G-matrices relate to each other, a set of four operators named circulant operators is introduced. These operators provide a new insight into the structure of the space of square matrices.

Key–Words: Pascal Matrices, Pascal Triangle, Circulant Operators, Square Matrix bipartition, cobweb Partition, G-matrices,

1 Introduction Deﬁnition 5 : G5,n

This section presents the definition of the twelve G- n − i matrices, G k as in [2]. In the sequel, [G5,n] = i, j = 0, ..., n k,n 1 ≤ ≤ 12 ij n − j we write [Gk,n]ij to denote the coefficient at the inter- section of row i and column j and denote by Gn the Definition 6 : G6,n set of the twelve G-matrices. Generic binomial coefficients a! will be denoted a . The Pascal triangle 2n − i − j (a−b)!b! b if i + j ≥ n is assumed to comprise levels 0 . . . n. [G6,n]ij =  n − j  0 otherwise Definition 1 : G 1,n  Definition 7 : G7,n i if i ≤ j [G ] =  j n − j 1,n ij [G ] = i, j = 0, ..., n  0 otherwise 7,n ij n − i  Definition 2 : G2,n Definition 8 : G8,n j n − j [G ] = i, j = 0, ..., n [G8,n] = i, j = 0, ..., n 2,n ij n − i ij i

Deﬁnition 3 : G3,n Deﬁnition 9 : G9,n

n + j − i n + i − j if i ≥ j if i ≤ j [G3,n]ij =  j [G9,n]ij =  i  0 otherwise  0 otherwise   Deﬁnition 4 : G4,n Deﬁnition 10 : G10,n

n − i i [G ] = i, j = 0, ..., n [G ] = i, j = 0, ..., n 4,n ij j 10,n ij n − j Proceedings of the 10th WSEAS International Confenrence on APPLIED MATHEMATICS, Dallas, Texas, USA, November 1-3, 2006 221

Deﬁnition 11 : G11,n the generic matrix subscript vector (i, j) (0 ≤ i, j ≤ n). i [G ] = i, j = 0, ..., n 11,n ij j Deﬁnition 13 The α-circulant operator

Definition 12 : G12,n i , j −−−→α i , i + j mod n + 1 i + j if i + j ≤ n [G12,n]ij =  i Definition 14 The β-circulant operator  0 otherwise  β Matrices like the above and other matrix forms de- i , j −−−→ −1 + i − j , j mod n + 1 rived from the Pascal triangle are encountered in [3, 9, 10, 1, 4, 16, 8]. Every author simply refers to Definition 15 The γ-circulant operator the particular form encountered as the Pascal matrix. γ Some authors refer to G1, n and G11 ,n respectively i , j −−−→ i − j , j mod n + 1 as upper and lower triangular Pascal matrices [5]. In [7, 8], the fourth G-matrix, G4, n is called binomial matrix. But, as it can be seen from the above defini- Definition 16 The δ-circulant operator tions, this is just one of the three possible upper-left δ triangular forms. In [11] one recognizes G12, n, the i , j −−−→ i , 1 + i + j mod n + 1 third northwest triangular matrix form. [12] refers to G7, n as the reflection of G11, n about the main antidiagonal. Considering the matrix formulation (1) of the well- known binomial theorem as proposed in [3],

1 (1 + t) (1 + t)2 ... (1 + t)n = 0 1 n Figure 1: The α-circulant operator 0 0 ...... 0  1 n  0 1 ...... 1  . . . .  As illustration, ﬁgure 1 shows how α transforms a 1 t t2 . . . tn  ......  (1)   square matrix.  . . . .   ......   . .  Circulant transformations carry circulant matrices    0 ...... 0 n  to associated row- or column-constant matrices. To  n  see this, consider the circulant matrix C given be- G1 , n low and its associated row-constant matrix denoted C˜: | {z } these matrices can be seen as some reordering of the components of the polynomial power basis vector, and c1 c2 c3 ··· cn hence of the polynomial space [3].  cn c1 c2 ··· cn−1  To our knowledge, this is the ﬁrst systematic attempt cn−1 cn c1 ··· cn−2 C =  , to investigate matrix forms of the Pascal triangle as  . . . . .   ......  mathematical objects in their own right.    c c c ··· c  Section 2 presents the set of four operators named cir-  2 3 4 1  culant operators and the way they intervene in gen- cn cn cn ··· cn erating the set of the twelve Pascal matrices, starting  cn−1 cn−1 cn−1 ··· cn−1  with one of them. Section 3 generalize to the case of ˜ c c c ··· c C =  n−2 n−2 n−2 n−2  any matrix.  . . . .   . . . .     c1 c1 c1 ··· c1  2 The circulant operators   This section is dedicated to the presentation of the set of circulant operators. It is readily seen for example that: C˜ = βC. These operators are presented as transformations of Proceedings of the 10th WSEAS International Confenrence on APPLIED MATHEMATICS, Dallas, Texas, USA, November 1-3, 2006 222

Theorem 17 Action of circulant operators on Gn NW/se NE/sw

If i ∈ {1, 5, 9} then βGi,n = Gi+1 [12],n If i ∈ {2, 6, 10} then δGi,n = Gi+1 [12],n If i ∈ {3, 7, 11} then γGi,n = Gi+1 [12],n If i ∈ {4, 8, 12} then αG = G i,n i+1 [12],n SW/ne SE/nw where i + 1 [12] ≡ i + 1 mod 12. Figure 2: Four bipartitions of a square matrix The stage is now set to derive all the twelve G- matrices starting with any particular one. Thus starting with G1,n, the twelve G-matrices can be derived in their natural order by the following sequence: deﬁne the North-East/south-west bipartition of A. Symbolically, we write:

β δ γ α G1,n −−−→ G2,n −−−→ G3,n −−−→ G4,n −−−→ G5,n A = ANE + Asw ≡ ANE/sw

β δ γ α Definition 19 The South-West/north-east bipartition G5,n −−−→ G6,n −−−→ G7,n −−−→ G8,n −−−→ G9,n Let A be a (n + 1) × (n + 1) matrix with subscripts β δ γ α i and j ranging from 0 to n. The two matrices ASW G9,n −−−→ G10,n −−−→ G11,n −−−→ G12,n −−−→ G1,n and Ane: [A] if i ≥ j [A ] = ij SW ij 0 if i < j 3 This is a cycle since (αγδβ) G1,n = G1,n. Moreover, and [A]ij if i < j it is readily verified that {G1,n,G5,n,G9,n} is glob- [Ane]ij = ally (αγδβ)-invariant. 0 if i ≥ j 3 define the South-West/north-east bipartition of A. Similar inspections show that: (γδβα) G4,n = G4,n, 3 3 Symbolically, we write: (δβαγ) G3,n = G3,n, (βαγδ) G2,n = G2,n, The circulant transformations provide a useful frame- work for investigating the G-matrices. Section 3 ex- A = ASW + Ane ≡ ASW/ne tends their application to square matrices to provide Definition 20 The North-West/south-east bipartition new insights into the structure of the space of square let A be a n n matrix with subscripts matrices. ( + 1) × ( + 1) i end j ranging from 0 to n. The two matrices ANW and Ase: [A] if i + j ≤ n 3 Applications and generalization [A ] = ij NW ij 0 if i + j > n 3.1 Triangular bipartition of a square matrix and [A] if i + j > n [A ] = ij Now, the four triangular bipartitions of any square se ij 0 if i + j ≤ n matrix are illustrated in figure 2. The triangular define the North-West/south-east bipartition of A. sub-block in full line indicates the block embracing Symbolically, we write: the entries of the main- or antidiagonal. A = ANW + Ase ≡ ANW/se

Deﬁnition 18 The North-East/south-west bipartition Deﬁnition 21 The South-East/north-west bipartition Let A be a (n + 1) × (n + 1) matrix with subscripts Let A be a (n + 1) × (n + 1) matrix with subscripts i and j ranging from 0 to n. The matrices two ANE i end j ranging from 0 to n. The matrices ASE and and Asw: Anw, such that: [A] if i ≤ j [A] if i + j ≥ n [A ] = ij [A ] = ij NE ij 0 if i > j SE ij 0 if i + j < n and and [A] if i > j [A] if i + j < n [A ] = ij [A ] = ij sw ij 0 if i ≤ j nw ij 0 if i + j ≥ n Proceedings of the 10th WSEAS International Confenrence on APPLIED MATHEMATICS, Dallas, Texas, USA, November 1-3, 2006 223

Subsets Matrices Subsets Matrices NE/sw NE/sw Gn G1,n G5,n G9,n An A1,n A5,n A9,n SE/nw G G G SE/nw Gn 2,n 6,n 10,n An A2,n A6,n A10,n SW/ne Gn G3,n G7,n G11,n SW/ne An A3,n A7,n A11,n NW/se Gn G4,n G8,n G12,n NW/se An A4,n A8,n A12,n

Table 1: Partition of the set Gn Table 2: Partition of the set An

deﬁne the South-East/north-west bipartition of A. Symbolically, we write: OAi,n = Ai+1 [12],n and A1,n = A with:

A = ASE + Anw ≡ ASE/nw i + 1 [12] = i + 1 modulus 12, β if i ∈ {1, 5, 9} 3.2 Partition of the set G induced by the cir-  δ if i ∈ {2, 6, 10} n and O =  culant operators and generalization  γ if i ∈ {3, 7, 11} α if i ∈ {4, 8, 12} As it can be seen, each of the twelves G-matrices  In ﬁne:  presents a natural bipartition: a major sub-block with non-zero entries and a minor sub-block with zero en- A β δ γ α tries. 1,n −−−→ A2,n −−−→ A3,n −−−→ A4,n −−−→ A5,n This leads to the natural partition of Gn as presented β δ γ α in Table 1. A5,n −−−→ A6,n −−−→ A7,n −−−→ A8,n −−−→ A9,n NE/sw It is easily veriﬁed that the set Gn is invariant β δ γ α (globally) under the action of αγδβ. A9,n −−−→ A10,n −−−→ A11,n −−−→ A12,n −−−→ A1,n SE/nw SW/ne NW/se So are Gn , Gn and Gn respectively under βαγδ, δβαγ and γδβα. This leads to the set An of twelve new matrices from It follows that in additions, it can be shown that: the single matrix A. Figure 4 shows a graph with NE/sw β SE/nw δ SW/ne γ An as set of vertices, and the circulant operators as Gn −−−→ Gn −−−→ Gn −−−→ NW/se α NE/sw transitions labels. Gn −−−→ Gn

A α 4,n A5,n A3,n

A6,n A2,n

A A γ β 7,n 1,n

A8,n A12,n

A9,n A11,n

A10,n δ Figure 4: Ring behavior of An with alphabet in Figure 3: Four bipartitions linked by circulant opera- {α, β, γ, δ} tors

O In what precedes, −−−−→ denotes the transformation by the operator O as shown in ﬁgure 3 which gives 4 New structure of the space of the generalization to the case of any square matrix A. Table 2 summarizes the general case by analogy to square matrices G . n Let A A be a square matrix: More explicitly, = [ ]i,j 0≤i,j≤n - the set of the coefﬁcients of A will be denoted by Proceedings of the 10th WSEAS International Confenrence on APPLIED MATHEMATICS, Dallas, Texas, USA, November 1-3, 2006 224

Coef(A), To highlight one way in which the G-matrices relate to - the permutation group on {(i, j), 0 ≤ i, j ≤ n} will each other, the set of circulant operators is introduce. be denoted by P erm(n), These operators turn out to provide a new insight into - For σ ∈ P erm(n): the structure of the space of square matrices. It is es- σ((i, j)) ≡ (iσ , jσ) tablish that, beginning with a single matrix, one can A σA , i, j n, derive in this space concentric orbits describing a cob- [ ]iσ,jσ ≡ [ ]i,j 0 ≤ ≤ - σAn = {σA, A ∈ An} web partition graph. Exploring the cobweb cyclically, - Ak,n = {σAk,n, σ ∈ P erm (n)} where 1 ≤ k ≤ one can reach the twelve matrices (which can be seen 12 as state of a particular system) through the four bipartitions. On the other hand, a radial trajectory enable to Definition 22 A permutation σ ∈ P erm(n) pre- access ”states” in which the two sets of coefficients in serves a partition AB/b if there exists σB and σb such the bipartition are globally. that σ = σB σb Acknowledgements: The research was supported by where : the project STICO of ICD-FRE CNRS 2848 (Univer- sity of Technology of Troyes - UTT) in the case of the [Ab]i ,j = [Ab]i,j ∀ i, j σB σB first author. [AB] = [AB] ∀ i, j iσb ,jσb i,j In the sequel this property will be symbolized by the identity: σ AB/b = AB/b. References: [1] G. S. Call; Daniel J. Velleman, Pascal’s Matri- ces, The American Mathematical Mounthly,100, Theorem 23 Partition of σAn 4, 1993, pp. 372-376 If σ ∈ P erm(n) and σ ANE/sw = ANE/sw then [2] P. K. Doh, Twelve Matrix Form of the Arith- 12 metic Triangle: Mathematical Tools, Relation σAn = Ak,n and Ar,n ∩ As,n = ∅ ∀ r 6= s Diagrams, Personnal Communications, 2004, k[=1 [3] P. K. Doh, Courbes parametriques´ polynomiales Proof: et formes matricielles du theor´ eme` binomial: It follows directly from the definition of the set Ar,n. Nouveaux outils fondamentaux pour la concep- ⊓⊔ tion et fabrication assiste par ordinateur, Thesis of University of Nancy 1, 1988 The above theorem leads to a partitioning of the set [4] A. Edelman and G. Strang, Pascal’s Matrices, into orbits σ as represented in figure 5. An Ar,n The American Mathematical Mounthly, 111, 3,

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