Packing polynomials on multidimensional integer sectors

Transcription

1 Pacing polynomials on multidimensional integer sectors Luis B Morales IIMAS, Universidad Nacional Autónoma de México, Ciudad de México, 04510, México lbm@unammx Submitted: Jun 3, 015; Accepted: Sep 8, 016; Publised: Oct 14, 016 Matematics Subject Classifications: 11C08, 05A10, 05A19, 06R05 Abstract Denoting te real numbers and te nonnegative integers, respectively, by R and N, let S be a subset of N n for n 1,,, and f be a mapping from R n into R We call f a pacing function on S if te restriction f S is a bijection onto N For all positive integers r 1,, r n 1, we consider te integer sector I(r 1,, r n 1 {(x 1,, x n N n x i+1 r i x i for i 1,, n 1} Recently, Melvyn B Natanson (014 proved tat for n tere exist two quadratic pacing polynomials on te sector I(r Here, for n > we construct n 1 pacing polynomials on multidimensional integer sectors In particular, for eac pacing polynomial on N n we construct a pacing polynomial on te sector I(1,, 1 Keywords: Pacing polynomials; s-diagonal polynomials; multidimensional lattice point enumeration 1 Introduction In tis paper, N and R denote, respectively, nonnegative integers and real numbers, 0 < n N, and let S be a subset of N n A function f from R n into R is a pacing function on S if te restriction f S is a bijection onto N Also s(x x x n wen x (x 1,, x n N n Given w in N, let H(n, w {x N s(x w} A pacing function f on N n is called a diagonal mapping if f taes H(n, w bijectively onto {0, 1,, 1+ ( n+w n } for eac w N, or equivalently, if f(x < f(y wenever x, y N n and s(x < s(y (see [4, 5] Pacing functions map arbitrarily large n-dimensional arrays into computer memory cells numbered 0, 1,, and produce no conflicts in suc a process (see [7] Let n > 1 For all positive real numbers α 1,, α n 1, we define te real sector S(α 1,, α n 1 {(x 1,, x n R n 0 x i+1 α i x i, i 1,, n 1} te electronic journal of combinatorics 3(4 (016, #P45 1

2 and te integer sector and I(α 1,, α n 1 {(x 1,, x n N n 0 x i+1 α i x i, i 1,, n 1} We also define, respectively, te real and integer sectors S(,, {(x 1,, x n R n x i 0, i 1,, n} I(,, {(x 1,, x n N n x i 0, i 1,, n} N n Given any permutation π on {1,, n} and any n-tuple x (x 1,, x n R n, define πx (x π(1,, x π(n Ten, we say tat two functions f and g on R n are equivalent if tere exists a permutation π suc tat for all x, f(x g(πx Hereafter, we will write permutations in cycle notation It is not ard to see tat if (x 1,, x n is a variable vector and is a positive integer, ten te binomial coefficient ( +x 1 + +x n produces a degree polynomial in te variables x 1,, x n Pacing functions were introduced to literature by Caucy [3] Later, Cantor [1, ] observed tat te polynomial function ( x1 + x + 1 f(x 1, x + x (1 is a bijection from N onto N, ence a pacing function on I(, in our terms Te polynomials f and fπ, wit π (1, are called te Cantor polynomials More generally, Solem [?,?] constructed just one inequivalent n-dimensional pacing polynomial on I(,, for eac n > 0 Morales and Lew [5] constructed n inequivalent pacing polynomials of dimension n for eac n > 1 Morales [4] produced a family of pacing polynomials, wic includes te Morales-Lew polynomials Finally, Sancez [8] obtained a family of (n 1! inequivalent pacing polynomials on I(,,, wic includes te above family All tese polynomials are diagonal functions Recently Natanson [6] proved tat for n tere exist two quadratic pacing polynomials on te sector I(r Also e proved tat pacing polynomials on I( are in bijection wit pacing polynomials on I(1 For n > 1 we now construct n 1 pacing polynomials a generalization of Natanson s on I(r 1, r n 1, suc tat r 1,, r n 1 are positive integers Also, we prove tat tese n 1 pacing polynomials ave a ind of diagonal property In particular, for eac pacing polynomial on N n we construct a pacing polynomial on I(1,, 1 s-diagonal functions on integer sectors In tis section we construct recursive subsets of multidimensional integer sectors Using tese sets we define s-diagonal pacing functions on multidimensional integer sectors te electronic journal of combinatorics 3(4 (016, #P45

3 Tese functions are a generalization of te Natanson s functions Also we calculate te cardinalities of tese sets, wic are used to construct s-diagonal pacing functions Given any nonnegative integer x, we define For all positive integers r, we define E(1, x {x} E r (, x {(x, x N x E(1, x, x 0,, rx} For n > and all positive integers r 1, r,, r n 1, E r1,r,,r n 1 (n, x {(x, x,, x n N n It is not difficult to verify tat if x x, ten (x,, x n E r,,r n 1 (n 1, x, x 0,, r 1 x} ( E r1,r,,r n 1 (n, x E r1,r,,r n 1 (n, x (3 Note tat our set E r (, x coincides wit te set defined in [6, Teorem 7] Moreover, if r 1, r are positive integers, by direct calculation we get E r1,r (3, x {(x, 0, 0, (x, 1, 0,, (x, 1, 1, r,, (x, r 1 x, 0,, (x, r 1 x, r 1 r x} Given any positive integers r 1, r,, r n 1, and any nonnegative integer x, we define D r1,r,,r n 1 (n, x x E r1,r,,r n 1 (n, j (4 It is easy to see tat if r 1, r,, r n 1 are positive integers, ten I(r 1,, r n 1 x N E r1,r,,r n 1 (n, x Moreover, from ( and (4 we ave E r1,r,,r n 1 (n, x {(x, x,, x n N n (x,, x n D r,,r n 1 (n 1, r 1 x} (5 for any x N For use as te basis of an induction below, we define te 1-dimensional integer sector, I 1, as te set of nonnegative integers Let r be a positive integer Ten Natanson s [6] pacing polynomials on I(r ave a special property: for eac x N, any Natanson s polynomial maps D r (, x (resp E r (, x bijectively onto {0,, 1 + D r (, x } (resp {0, 1,, 1 + E r (, x } In analogy wit te definition of diagonal functions on N, a pacing function f on I(r is called an s-diagonal function if f as tis property We generalize tis definition for pacing functions on multidimensional integer sectors as follows te electronic journal of combinatorics 3(4 (016, #P45 3

4 Let r 1, r,, r n 1 be positive integers A pacing function f on I(r 1,, r n 1 is called a s-diagonal mapping if f maps D r1,r,,r n 1 (n, x bijectively onto {0,, 1 + D r1,r,,r n 1 (n, x } Also for eac n > 0, we define s-db r1,,r n 1 (n and s-dp r1,,r n 1 (n to be te sets of s-diagonal functions and s-diagonal polynomials defined on R n, respectively (s-db 1 (1 and s-dp 1 (1 denote te s-diagonal functions and s-diagonal polynomials on I 1 Tese definitions clearly imply te following lemma Lemma 1 If r 1, r,, r n 1 are positive integers, ten (1 If f DB r1,,r n 1 (n, ten f(0,, 0 0 ( f DB r1,,r n 1 (n if and only if f(x 1, x,, x n < f(y 1, y,, y n wenever (x 1, x,, x n, (y 1, y,, y n I(r 1,, r n 1 and x 1 < y 1 (3 If f DB 1 (1, ten f is te identity map on I 1 Now we calculate te cardinalities of te sets E r1,r,,r n 1 (n, x and D r1,r,,r n 1 (n, x Denote te cardinalities of te sets E r1,r,,r n 1 (n, x and D r1,r,,r n 1 (n, x by te functions T r1,r,,r n 1 (x and Q r1,r,,r n 1 (x, respectively Also we define T r1,r,,r n 1 ( 1 Q r1,r,,r n 1 ( 1 0 Te next lemma calculates te cardinality of tese sets Lemma If r 1, r,, r n 1 are positive integers and x is any nonnegative integer, ten T r1,r,,r n 1 (x Q r1,r,,r n 1 (x i 0 r n 1 i x r 1 i 1 i 1 0 i 0 i n0 r n 1 i i n0 1, (6 1, (7 Q r1,r,,r n 1 (x Q r1,r,,r n 1 (x 1 + Q r,,r n 1 (r 1 x (8 Proof Clearly, E(1, x 1 For n > 1, relation (3 and te induction ypotesis imply tat r 1 x r j T r1,r,,r n 1 (x T r,,r n 1 (j Tus (6 olds From (3, (4 and (6 we obtain Q r1,r,,r n 1 (x D r1,r,,r n 1 (n, x Tus (7 olds x r 1 i 1 i 1 0 i 0 r n 1 i i n0 1 i 3 0 r n 1 i i n0 x E r1,r,,r n 1 (n, i 1 i x T r1,r,,r n 1 (i 1 i 1 0 te electronic journal of combinatorics 3(4 (016, #P45 4

5 By (7 we ave tat Q r1,r,,r n 1 (x Tis completes te proof x r 1 i 1 i 1 0 i 0 r n 1 i i n x 1 r 1 i 1 i 1 0 i 0 r n i n Q r1,r,,r n 1 (x 1 + Q r,,r n 1 (r 1 x Using binomial coefficient identities, we can cec tat ( n 1 + x T 1,,1 (x, n 1 ( n + x Q 1,,1 (x n i n r i i 0 i 3 0 r n 1 i i n0 1 3 Polynomial formula In tis section we express te functions T r1,r,,r n 1 (x and Q r1,r,,r n 1 (x as polynomials in te variable x In order to prove tis, for any positive integer r we first construct a family of polynomials in te variable x, { ( } x P r, (x c r,s 1,, n, s wose coefficients c r,j c r,j j+1 s1 s1 satisfy te recurrence relations ( r c r s,j 1 for > j wit c r,1 s ( r + 1 and c r r (9 (wen no confusion arises we omit te superscript r Since c r and c, rc 1, 1 for, it follows tat c, r for 1 Example For n 3, we ave ( ( i r + 1 P r,1 (i ri, P r, (i r + i, P r,3 (i r 3 ( i 3 ( ( i r r Let r and i be two positive integers and t be any integer suc tat 0 t < r It is not ard to cec tat for every positive integer, r ( ( j r + 1, ( ri+t ( j ri ( j ri i 1 ( j + j1 i t ( ir + j, (11 r ( rj + l l1 i 1 r l1 rj+l 1 s0 ( s (1 1 te electronic journal of combinatorics 3(4 (016, #P45 5

6 Tese binomial coefficient identities will be used in te proof of te following lemma Lemma 3 Let r be a positive integer Ten, for any positive integer i and any integer 0 t < r, ri+t ( j P r,+1 (i + 1 ( t P r,+1 (i + ( t Proof We give an inductive proof on Assume tat 1 Ten by (10-(1 we obtain ri+t ( j 1 ( j t ( ri + j i 1 r + [rj + s] j1 s1 i 1 [ ( ] ( r + 1 t + 1 r j + + tri + ( ( ( i r + 1 t + 1 r + i + tri + ri t [ri + j] Since ( i r + ( r+1 i Pr, (i and ir P r,1 (i, te formula olds for 1 Now suppose tat te induction ypotesis is true for 1 Note tat (11 allows us to divide te proof in two parts: Part A Here we prove tat ri ( j Pr,+1 (i From (10 and (1 we obtain ri ( j i 1 1 r ( rj + l l1 i 1 Ten, by induction ypotesis ri ( [ j i 1 r 1 ( l 1 P r, (j + P r, (j + l1 1 [ i 1 1 ( r rp r, (j + P r, (j [ i 1 s1 ( ( r j c,s 1 s s1 [ i 1 ( ( r j 1 c +1, ( ] r r l1 rj+l 1 m0 ( ] l ( ] r j1 ( m 1 ( ( r j c,s s ( ( r j c +1, + + ( ] r ( ( r j c, 1 te electronic journal of combinatorics 3(4 (016, #P45 6

7 [ i 1 s1 s1 s1 +1 s 1 +1 s 1 +1 s ( ( r j c +1,s + s 1 i 1 c +1,s ( r ( j i 1 + s ( ] r ( r ( ( ( r i r + 1 c +1,s + i s ( ( ( r i r + 1 c +1,s 1 + i s s+1 s 1 However, from (9 we get c +1,s +1 s+1 ( r 1 c+1,s 1 for + 1 > s Terefore ri ( j +1 s c +1,s ( i s + ( r + 1 i P r,+1 (i + 1 Tus we ave proved part A Part B Here we prove tat t ( ir+j j1 ( t 1 Pr,+1 (i + ( t+1 +1 By (10 and (1 we get t ( ir + j j1 1 t j1 ir+j 1 s1 ( s 1 Ten, by induction ypotesis t ( [ ir + j t 1 ( j 1 P r, (i + j1 j1 1 1 ( t tp r, (i + P r, (i ( t P r,+1 (i + Tus we ave proved part B Finally, parts A, B and (11 imply te lemma P r, (i + ( t ( ] j ( t In te following teorem we sow tat te functions T r1,r,,r n 1 (x and Q r1,r,,r n 1 (x can be extended to polynomials in te variable x For use as te basis of an induction below, since E(1, x {x} and D(1, x {0,, x}, we can define T 1 (x 1 and Q 1 (x x + 1 te electronic journal of combinatorics 3(4 (016, #P45 7

8 Teorem 4 Let r 1, r,, r n 1 be positive integers Ten T r1,r,,r n 1 (x c r n 1 Q r1,r,,r n 1 (x c r n 1 i 1 c r n,i i +1 i 3 1 +T r1,,r n (x, i +1 i 1 c r n,i +Q r1,,r n (x i 3 1 i n +1 i +1,i 3 i n 1 1 i n +1 i +1,i 3 i n 1 1 ( c r 1 x i n +1,i n 1 i n 1 ( c r 1 x + 1 i n +1,i n 1 i n (13 (14 Moreover, bot functions T r1,r,,r n 1 (x and Q r1,r,,r n 1 (x are polynomials in x of degree n 1 and n, respectively Proof From Lemma we get T r1 (x i r 1 x + 1 c r 1 ( x 1 + T 1 (x, Q r1 (x c r 1 ( x x + 1 Tus te lemma olds for n Now suppose tat n > It follows from Lemma and te induction ypotesis tat T r1,r,,r n 1 (x r j j 0 j 3 0 j 0 c r n 1 r n 1 j i 1 j n0 c r n,i + T r,,r n (j j 0 c r n 1 i 1 c r n,i i +1 i i +1 i 3 1 j 0 T r,,r n 1 (j i n 3 +1 i +1,i 3 i n 3 +1 i +1,i 3 i n 1 i n 1 ( c r j i n 3 +1,i n i n c r i n 3 +1,i n j 0 ( j i n + T r,,r n (j (15 j 0 However, from Lemma 3 we get j 0 ( j i n P r1,i n +1(x i n +1 i n 1 1 ( c r 1 x i n +1,i n 1 i n 1 te electronic journal of combinatorics 3(4 (016, #P45 8

9 Using te above identity in (15 we obtain c r n 1 i 1 c r n 1 c r n,i i 1 i +1 i 3 1 c r n,i i n 3 +1 i +1,i 3 i +1 i 3 1 i n 1 c r i n 3 +1,i n i n 3 +1 i +1,i 3 On te oter and, from (6 we ave j 0 T r,,r n (j r j j 0 i 3 0 i n 1 r n i n j 0 c r i n 3 +1,i n i n 1 0 ( j i n i n +1 i n 1 1 ( c r 1 x i n +1,i n 1 (16 i n 1 1 T r1,r,,r n (x (17 Ten (15, (16 and (17 imply relation (13 Clearly, Equation (16 gives a polynomial in x Moreover, by induction ypotesis T r1,r,,r n (x is also a polynomial in x Terefore T r1,r,,r n (x is a polynomial in x From (13 it is clear tat i n 1 n 1; ence te degree of tis polynomial is n 1 Now we sow (14 From (6 and (7 we deduce tat Q r1,,r m 1 (x x y0 T r 1,,r n 1 (y Ten Using (13, we can see tat Q r1,,r n 1 (x x y1 c r n 1 c r n 1 c r n 1 i 1 i 1 i 1 c r n,i c r n,i c r n,i i +1 i 3 1 i +1 i 3 1 i +1 i 3 1 i n +1 i +1,i 3 i n +1 i +1,i 3 i n 1 1 i n +1 i +1,i 3 i n 1 1 i n 1 1 ( c r 1 y i n +1,i n 1 i n 1 c r 1 i n +1,i n 1 x [( y y1 i n 1 c r 1 i n +1,i n 1 ( x + 1 i n T r1,,r n (y ] + T r1,,r n (y + Q r1,,r n (x In a similar way as in te proof of (13, we can sow tat Q r1,,r n 1 (x is a polynomial in x of degree n 4 s-diagonal polynomials on integer sectors Here, for n > 1 we construct recursive n 1 s-diagonal pacing polynomials on multidimensional integer sectors, using te polynomial formula of te function Q r1,r,,r n 1 Te recursive construction begins wit te s-diagonal identity polynomial on te integer sector I 1 Let r 1, r,, r n 1 be any positive integers and f be a real-valued function on N n 1 We define two operators F and G tat transform te function f into two functions on N n, as follows F f(x 1,, x n Q r1,r,,r n 1 (x f(x,, x n, (18 Gf(x 1,, x n Q r1,r,,r n 1 (x 1 1 f(x,, x n (19 te electronic journal of combinatorics 3(4 (016, #P45 9

10 Let r be a positive integer By Lemma 1(3, if I DB 1 (1, ten I is te identity map on I 1 (N Tus by direct calculation, F I and GI are te same Natanson polynomials defined in [6, Teorem 7] Also [6, Teorem 7] yields tat F I, GI DB r ( Teorem 5 Let r 1, r,, r n 1 be positive integers If n > 1 and f DB r,,r n 1 (n 1, ten F f, Gf DB r1,r,,r n 1 (n Proof To prove te teorem, we use a double induction on n and x 1 For n and x 1 arbitrary te statement follows from te preceding remar By definition, E r1,,r n 1 (n, 0 {(0,, 0}, so (4 implies tat D r1,,r n 1 (n, 0 {(0,, 0} Ten te result is true for all n > and x 1 0 Now assume tat n > and x 1 > 0 Ten by induction ypotesis, F f and Gf are bijections from D r1,,r n 1 (n, x 1 1 onto {0, 1,, 1 + Q r1,,r n 1 (x 1 1} However, by (3 and (4 we ave tat D r1,,r n 1 (n, x 1 is te disjoint union of D r1,,r n 1 (n, x 1 1 and E r1,r,,r n 1 (n, x 1 Hence to complete te proof of te teorem, we need only sow tat F f and Gf bijectively map E r1,r,,r n 1 (n, x 1 onto {Q r1,r,,r n 1 (x 1 1,, 1 + Q r1,r,,r n 1 (x 1 } On te oter, by te ypotesis, f bijectively maps D r,,r n 1 (n 1, r 1 x 1 onto {0, 1,, 1 + Q r,,r n 1 (r 1 x 1 } Ten it is easy to see tat Q r1,r,,r n 1 (x f and Q r1,r,,r n 1 (x 1 1 f bijectively map D r,,r n 1 (n 1, r 1 x 1 onto {Q r1,r,,r n 1 (x 1 1,, 1 + Q r1,r,,r n 1 (x Q r,,r n 1 (r 1 x 1 } and (0 { Q r,,r n 1 (r 1 x 1 + Q r1,r,,r n 1 (x 1,, 1 + Q r1,r,,r n 1 (x 1 }, (1 respectively By (8 we get Q r1,r,,r n 1 (x 1 Q r1,r,,r n 1 (x Q r,,r n 1 (r 1 x 1, ( Q r1,r,,r n 1 (x 1 1 Q r,,r n 1 (r 1 x 1 + Q r1,r,,r n 1 (x 1 (3 Ten (0, (1, ( and (3 imply tat Q r1,r,,r n 1 (x 1 1+f and Q r1,r,,r n 1 (x 1 1 f are bijections from D r,,r n 1 (n 1, r 1 x 1 onto {Q r1,r,,r n 1 (x 1 1,, 1 + Q r1,r,,r n 1 (x 1 } (4 However, by (5, (18, (19 and (4 we deduce tat bot Af and Bf bijectively map E r1,r,,r n 1 (n, x 1 onto {Q r1,r,,r n 1 (x 1 1,, 1 + Q r1,r,,r n 1 (x 1 } Tis completes te proof Teorem 6 Let r 1, r,, r n 1 be positive integers If n > 1 and f DP r,,r n 1 (n 1 ten F f(1, 0,, 0 1, Gf(1, 0,, 0 Q r1,,r n 1 ( 1 (5 Moreover, if n > 1 ten F f and Gf are distinct functions te electronic journal of combinatorics 3(4 (016, #P45 10

11 Proof From Teorems 4 and 5 we ave tat F f, Gf DP r1,r,,r n 1 (n Lemma 1, (18 and (19 yield relations (5 By definition, if n > 1, ten Q r1,r,,r n 1 ( D r1,r,,r n 1 (n, 1 3 Tis result and (5 imply te last statement Given any positive integers r 1, r,, r n 1 wit n > 0, ten by Teorems 4 and 5 we can define a family of s-diagonal degree n polynomials, QDr 1, r,, r n 1 (n, on I(r 1,, r n 1 suc tat QD 1 (1 DP 1 (1, and for n > 1, QD r1,r,,r n 1 (n {F f, Gf f QD r,,r n 1 (n 1 } Note tat if f and g are two different functions in DP r,,r n 1 (n 1, ten by (5, F f Gf Terefore, it follows from Teorem 6 tat QD r1,r,,r n 1 (n n 1 Example Let r 1 and r be any nonnegative integers and I DB 1 (1 By direct calculation, te four s-diagonal polynomials of te set QD r1,r (3 are ( ( ( ( F F I(x 1, x, x 3 r r1 x1 r1 + 1 x1 x1 + r + r 1 3 ( ( ( ( F GI(x 1, x, x 3 r r1 x1 r1 + 1 x1 x1 + r + r x 3, ( ( ( GF I(x 1, x, x 3 r r1 x1 + 1 r1 + 1 x r 3 [ ( ] x r + x + x 3, ( ( ( GGI(x 1, x, x 3 r r1 x1 + 1 r1 + 1 x r 3 [ ( ] x + 1 r + x x 3 + x 1 + r ( x + x 1 + r ( x x + x 3, + x ( x r 1 + x ( x r 1 + x s-diagonal polynomials on I(1,, 1 In tis section we study te relation between pacing polynomials on I(1,, 1 and on I(,, Here I denotes te identity map on N Let f be a real-valued function on N n 1 Morales and Lew [5] defined te operators A and B tat transform te function f into two functions on N n, as follows ( n 1 + x1 + + x n Af(x 1,, x n + f(x,, x n, (6 n ( n + x1 + + x n Bf(x 1,, x n 1 f(x n 1,, x 1 (7 n Tey proved tat if f is a diagonal polynomial on N n 1, ten bot Af and Bf are diagonal polynomials on N n In particular, tey proved tat bot AI and BI are te same te electronic journal of combinatorics 3(4 (016, #P45 11

12 Cantor polynomial (1 Tus tey constructed n inequivalent diagonal polynomials on I(,, For any positive n > 1, we define a linear transformation from R n to R n wose matrix wit respect to te standard base is Λ n It is easy to see tat Λ 1 n Teorem 7 If f is a diagonal polynomial on I(,, ten f Λ 1 n is an s-diagonal polynomial on I(1,, 1 Moreover, if f is an s-diagonal polynomial on I(1,, 1, ten f Λ n is a diagonal polynomial on I(,, Proof Clearly Λ 1 n is a bijection from I(1,, 1 onto I(,, Let x (x 1,, x n, y (y 1,, y n I(,, Ten x 1 < y 1 if and only if s(λ 1 n (x < s(λ 1 n (x Terefore, if f is a diagonal polynomial on I(,,, ten f Λ 1 n is an s-diagonal polynomial on I(1,, 1 Te last statement is proved in a similar way Example Te four s-diagonal polynomials on I(1, 1 satisfy F F I(x 1, x, x 3 AAIΛ 1 (x 1, x, x 3, F GI(x 1, x, x 3 AAI(1(3Λ 1 (x 1, x, x 3 GF I(x 1, x, x 3 BAI(13(Λ 1 (x 1, x, x 3, GGI(x 1, x, x 3 BAI(13Λ 1 (x 1, x, x 3 Tese identities are not ard to prove For example if (x 1, x, x 3 I(1, 1, ten BAI(13(Λ 1 (x 1, x, x 3 BAI(13((x 1 x, x x 3, x 3 BAI(x 3, x x 3, x 1 x ( 3 + x1 1 AI(x x 3, x 3 3 ( ( 3 + x1 1 + x 1 x 3 GF I(x 1, x, x 3 3 te electronic journal of combinatorics 3(4 (016, #P45 1

13 Acnowledgements Te autor would lie to tan anonymous referees for teir suggestions to improve te paper References [1] G Cantor Ein Beitrag zur mannigfaltigeitslere J Reine Angew Mat (Crelle s Journal, 84:4 58, 1878 [] G Cantor Beitrage zur begrundung der transfiniter mengenlere Mat Ann 46:481 51, 1885; translated as Contributions to te Founding of te Teory of Transfinite Numbers, Dover, New Yor, 195 [3] AL Caucy Cours d analyse de l École Royale Polytecnique, 1re partie: Analyse algébrique, 1 Imprimerie Royale, Paris 181, reprinted by WissenscFftlice BucgesellscFft, Darmstadt, pages , 1968 [4] LB Morales Diagonal polynomials and diagonal orders on multidimensional lattices Teory of Comput Systems, 30(4:367 38, 1997 [5] LB Morales, and JS Lew An enlarged of family pacing polynomials on multidimensional lattices Mat Systems Teory, 9(3:93 303, 1996 [6] MB Natanson Cantor polynomials for semigroup sectors J Algebra Appl 13( (14 pages, 014 [7] A L Rosenberg Storage mappings for extendible arrays Current Trends in Programming Metodology, Vol IV, (R T Ye ed, Prentice Hall, Englewood Cliffs, NJ, 1978, pp [8] A Sancez-Flores A family of (n 1! diagonal polynomials orders on N n Order, 1(: , 1997 te electronic journal of combinatorics 3(4 (016, #P45 13