Latin rectangles and quadrature formulas

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Latin rectangles and quadrature formulas N.M. Dobrovol’skii, N.N. Dobrovol’skii, I.Yu. Rebrova, I.N. Balaba Tula State Pedagogical University, Russian Federation

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Article history: Available online xxxx Dedicated to the memory of Mikhail Efimovich Desa

a b s t r a c t We consider the relationship between the classical combinatorial object — Latin rectangles and multidimensional quadrature formulas on the class Esα . © 2018 Elsevier Ltd. All rights reserved.

1. Introduction The problem of Latin rectangles was delivered by L. Euler in the middle of the XVIII century [19,21]. In this field, there were famous mathematicians such as Jacob [13], Kaplansky [15], Carlitz [1], Kerawala [18], Riordan [16] and Yamamoto [14]. In the summer of 1964, Mikhail Nikolayevich Dobrovolsky (27.10.1922–18.1.1975) obtained a solution of L. Euler’s problem of four-line Latin rectangles. A short communication was published in 1968 [6]. More detail with the works of M. N. Dobrovolsky are described in [2–11]. K. A. Rybnikov wrote in his book ([20], p. 115) ‘‘In 1965, at a seminar at the Moscow State University, M. N. Dobrovolsky from Tula reported on the number of N(4, n) found by him’’. This result was the main in the Ph.D. thesis of M. N. Dobrovolsky, which he defended on January 18, 1971 in Dissertational Council of Moscow State Pedagogical University named after V. I. Lenin on specialty 01.004. — Algebra and Number Theory.1 The scientific adviser was associate professor V. D. Podsypanin (10.1.1910–30.09.1968). The official opponents were Doctor of Physical and Mathematical Sciences, Professor N. Ya. Vilenkin (30.10.1921–19.10.1991) and Doctor of Physical and Mathematical Sciences, Professor K. A. Rybnikov (18.8.1913–20.8.2004). Vladimir State Pedagogical Institute was leading higher educational institution. On the defense instead of Professor K. A. Rybnikov as the second opponent was the Candidate of Physical and Mathematical Sciences M. E. Deza. Therefore, this topic has a very direct relation to Deza’s memory. The third issue of the volume 8 of Chebyshevskii Sbornik which was published in 2007 was dedicated to the 85th anniversary of the birth of Mikhail Nikolayevich Dobrovolsky. This collection E-mail address: [email protected] (N.M. Dobrovol’skii). 1 The specialty code and the title are given according to the specialty nomenclature existing at the time. https://doi.org/10.1016/j.ejc.2018.02.004 0195-6698/© 2018 Elsevier Ltd. All rights reserved.

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contains a lot of materials from the life and scientific activity of M. N. Dobrovolsky (senior), in which Mikhail Efimovich Deza played a significant role. In this paper we present a speech by M. N. Dobrovolsky on defense on January 18, 1971, where Mikhail Efimovich Deza was an official opponent. Unfortunately, we do not have the text of the speech of Mikhail Efimovich. Further we will discuss a new direction of research related to the development of Korobov’s number-theoretical method, which is the main for Tula School of Number Theory, founded in 1950 by the associate professor V. D. Podsypanin (see [12]). 2. Speech on thesis defense 01/18/1971 Dear comrades, The theme of my thesis is the solution of some combinatorial problems on permutations with constraint positions. One of the main problems of combinatorial theory is finding the number of objects of some set A given by the determining vector A(i1 , . . . , in ). A function of the set A which is equal to the number of objects of the set A is called the enumerator and denoted by N (A). The main method for solving such problems is the method of generating functions. We consider the function F (z) =

∑

N (A)z(A) =

Ω

∑

i

N (i1 , . . . , in )z11 . . . znin ,

Ω

where Ω is range of values of vectors A, and relations for the enumerators are transformed into relations for generating function. This method is very productive, since it allows us to involve all the power of the analysis apparatus in solving the problem. The main difficulties that remain for combinatorial theory are to determine the equations connecting the enumerators N (A). To find these equations, we apply a method that we will call the inclusion– exclusion method. Let A be the set of objects A and A1 be the set of objects A1 . Consider the procedure ω A ←→ A1 .

We say that the procedure ω is local if the number of transformations of the object of the set A in the procedure is completely determined by specifying the vector A and does not depend on the configuration of the object. If the number of transformations also depends on the configuration of the object, then such procedure will be called integral. The property of the procedure ω to be integral or local depends on the internal properties of objects of the set A. If the procedure ω is integral, we should try to decompose the set A into subsets so that the procedure ω becomes local. If the procedure is local, then we have equation

∑

f (A)N (A) =

A

∑

f −1 (A1 )N (A1 ),

A1

where f (A) is the number of transformations of the objects A in the procedure ω, and f −1 (A1 ) is the number of backward transformation of the objects A1 in the same procedure. The above considerations formed the basis for the solution of the following combinatorial problems on permutations with restriction of positions. I. On the number of permutations of elements n couple a1 , b1 ; . . . ;an , bn , in which the elements of l couple pairwise stand side by side, and the elements of k couple pairwise stand on the places of the same parity. For this problem A(n, l, k) is a determining vector, local procedure ω is inclusion–exclusion of a couple in which elements stand side by side and generating function is u(z , s, t) =

∞ ∑ n=0

n

n

n!z (1 − (s − 1)z)

−(2n+1)

[2] ∑

Cnk Cnk−k (2−1 t)2k

k=0

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in the case of a linear arrangement of the elements and

v (z , s, t) = −1 − 2sz + z + (1 + (s − 1)z)

∞ ∑

n

n

−(2n+1)

n!z (1 − (s − 1)z)

n=0

[2] ∑

Cnk Cnk−k (2−1 t)2k

k=0

in the case of cyclic arrangement of elements. II. On the number of three and four-line Latin rectangles. To build a local procedure, the problem is generalized and generalized Latin rectangles are considered, in which all elements are the same in m1 columns, and so on, in p columns, all the elements are different. Then the vector (m1 , n1 , n2 , n3 , k) is a determining vector for three-line generalized rectangles, and (m1 , n1 , n2 , n3 , n4 , k1 , k2 , k3 , l1 , l2 , l3 , l4 , l5 , l6 , p) is a determining vector for four-line generalized rectangles. The local procedures are inclusion–exclusion of a column consisting of identical elements, the transpositions of elements of top line, the transpositions of the elements of the first two lines, the transposition of the elements of the first three lines. The generating function is F (u) = e2u

∞ ∑

i!ui (1 + u)−3i−3

i=0

for three-line Latin rectangles and F (x) = e−6x

∞ ∑ ∞ ∑

(−1)β B(α, β )xα+β (1 + 2x)2α (1 − 2x)−(4α+2β+4) (1 − x)−(2α+β+3) ,

α=0 β=0

where (α + β1 + β2 + β3 )!(α + β1 + β4 + β5 )!(α + β2 + β4 + β6 )!

∑

B(α, β ) =

(α!)2 β1 !β2 !β3 !β4 !β5 !β6 !

β1 +···+β6 =β

·

· (α + β3 + β5 + β6 )! for four-line Latin rectangles. III. On the number of permutations with l neighbors. We say that there is one neighbor in the permutation, if the elements ai , ai+1 stand side by side. For this problem A(n, l) is a determining vector, the removal a1 is decision procedure. The generating function in the linear case is

Φ3 (n, t) =

n−1 ∑

(i + 1)!

i=0

T (n, i) + T (n − 1, i) 2

(t − 1)n−1−i ,

where T (n, i) is ith coefficient of expansion in powers of x of Chebyshev polynomial

( T (n, x) =

x+1+

√

(x + 1)2 + 4x 2n−1

)n

√

( )n √ − x + 1 − (x + 1)2 + 4x

(x + 1)2 + 4x

.

In the thesis, the tables of the enumerators are calculated for each of the three tasks and asymptotic formulas for the enumerators are received for the first two problems. 3. Latin rectangles and quadrature formulas Recall the definition of Latin rectangles.

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Definition 1. Latin rectangle is called a rectangular table consisting of n columns and s rows (n ≥ s), in each row of which there are n elements 1, 2, . . . , n such that in each row and every column all elements are different. Further, it is more convenient to use elements 0, 1, 2, . . . n − 2, n − 1 instead of 1, 2, . . . , n. Thus, each Latin rectangle after the column permutation ensuring the monotonicity of the elements of first row can be represented as a matrix n−2

n−1

a1,1

... ...

a1,n−2

a1,n−1 ⎟ ⎟

as−1,1

... ...

as−1,n−2

⎛

0

1

⎜ ⎜ ⎝

a1,0

.. .

L=⎜

.. .

as−1,0

.. .

.. .

⎞ ⎟, ⎠

as−1,n−1

satisfying the following conditions:

{aν,0 , aν,1 , . . . , aν,n−2 , aν,n−1 } = {0, 1, . . . , n − 2, n − 1} (ν = 1, . . . , s − 1),

(1)

(1 ≤ ν < µ ≤ s − 1, j = 0, . . . , n − 1),

aν,j ̸ = aµ,j

(2)

(1 ≤ ν ≤ s − 1, j = 0, . . . , n − 1).

aν,j ̸ = j

(3) s

Each matrix L can be associated with a net M consisting of n points of the unit cube Gs = [0; 1) by the following rule: M = {⃗ xj |j = 0, . . . , n − 1},

( ⃗xj =

j a1,j n

,

n

,...,

as−1,j

)

n

(4) (j = 0, . . . , n − 1).

(5)

The net M grid has a very important property. The projection of the net to any coordinate axis consists of n different points (see relation(1)). Recall that for arbitrary integers m1 ,. . ., ms the sums S(m1 ,. . ., ms ) defining by equality S(m1 ,. . ., ms ) =

N ∑

e2π i[m1 ξ1 (k)+···+ms ξs (k)] ,

(6)

k=1

is called trigonometric sums of the net M. Consider the simplest quadrature formula with M on the class Esα (α > 1): 1

∫

... 0

1

∫

f (⃗ x)d⃗ x= 0

n−1 1∑

n

f (⃗ xj ) − Rn [f ],

(7)

j=0

where Rn [f ] is linear functional of approximate integration error by quadrature formula with M. It is well known, that the class of functions Esα consists of periodic functions f (⃗ x) that have a rapidly convergent Fourier series:

∑

f (⃗ x) =

⃗ 2π i(m⃗ ,⃗x) , c(m)e

⃗ ∈Zs m

such that we have the following estimate for Fourier coefficients2 1

∫

...

⃗ = c(m)

1

∫

0

f (⃗ x)e

⃗ ,⃗x) −2π i(m

( d⃗ x=O

0

1 (m1 . . . ms )α

)

.

The class of periodic functions Esα by norm α ⃗ ∥f (⃗x)∥Esα = sup |c(m)(m 1 . . . ms ) |

(8)

⃗ ∈Zs m

2 Here and below, for real m we suppose m = max(1, |m|).

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is an inseparable Banach space, which isomorphic the space of all bounded sequences of complex numbers l0 . For linear functional of error of quadrature formula (7) the following theorem holds (see [17]). Theorem 1. If f (x1 , . . . , xs ) ∈ Esα , then for error of quadrature formula the following estimate holds:

|RN [f ]| ≤

∞ ∑ ′

∥f (⃗x)∥Esα N

m1

|S(m1 , . . . , ms )| , (m1 . . . ms )α ,...,m =−∞

(9)

s

where the sum S(m1 , . . . , ms ) defines by equality (6). This estimate cannot be improved on the class Esα . This estimate follows immediately from RN [f ] =

∞ ∑ ′

1 N

⃗ . S(m1 , . . . , ms )c(m)

m1 ,...,ms =−∞

Notice that the classical parallelepiped nets of Korobov M(a1 , . . . , as−1 ; N) = (aν , N) = 1

{(

k N

,

{

a1 k

}

N

,...,

{

} })⏐ ⏐ ⏐ k = 0, . . . , N − 1 , ⏐ N

as−1 k

(1 ≤ ν ≤ s − 1)

(10)

correspond to the matrix L of the form

⎛

0

⎜ ⎜0 ⎜ ⎜ L = ⎜. ⎜ .. ⎜ ⎝ 0

...

1 N

{a } 1

N

.. . N

{a

...

N

...

s−1

N

}

...

{ N −2 } a1 (N − 2)

{ N

N

.. .

as−1 (N − 2) N

}

⎞ { N −1 } a1 (N − 1) ⎟ ⎟ N ⎟ N ⎟ ⎟, .. ⎟ . { }⎟ as−1 (N − 2) ⎠

N

N

which, if you omit the null column, specifies s-line Latin rectangle with N − 1 columns for aν ̸ ≡ aµ (mod N) (1 ≤ ν < µ ≤ s − 1). Today we do not know how to estimate trigonometric sums corresponding to arbitrary Latin rectangles. But the following result in the manner of I. I. Pyatetskii-Shapiro is valid (see [17]). It is known, that Pyatetskii-Shapiro’s theorem stimulated N. M. Korobov to create a method of optimal coefficients which is one of the most important directions of the number-theoretic method in approximate analysis. Denote by L(n, s) the set of all matrices L, satisfying condition (1). It is clear that |L(n, s)| = (n!)s−1 . By M(R[f ]) we denote mathematical expectation of approximate integration error by quadrature formula corresponding to the matrix L ∈ L(n, s). Theorem 2. For any f (⃗ x) ∈ Esα the following estimate for mathematical expectation of approximate integration error

|M(R[f ])| ≤

s ∥f (⃗x)∥Esα ∑

nα

ν=1

Csν

2ν ζ ν (α ) nα (ν−1)

(11)

is valid.

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Proof. Let Sn be a symmetrical group of permutations of elements 0, 1, . . . . . . , n − 1, then arbitrary matrix L ∈ L(n, s) can be written in the form

⎛

0

... ...

1

⎜ π1 (0) ⎜ ⎜ . ⎝ .. πs−1 (0)

π1 (1) .. .

... ...

πs−1 (1)

n−2

n−1

⎞

π1 (n − 2) .. .

π1 (n − 1) .. .

⎟ ⎟ ⎟, ⎠

πs−1 (n − 2)

πs−1 (n − 1)

where π1 ,. . . ,πs−1 ∈ Sn . For the net M corresponding to the matrix L the trigonometric sum has the form S(m1 , . . . , ms ) =

n−1 ∑

e2π i

m1 k+m2 π1 (k)+···+ms πs−1 (k) n

.

k=0

It follows that 1

M(R[f ]) =

=

(n!

π1 ,...,πs−1 ∈Sn

∞ ∑ ′

1 n

)s−1

⃗ c(m)

m1 ,...,ms =−∞

∞ ∑ ′

1

∑ n−1 ∑

n

e2 π i

n−1 ∑

⃗ c(m)

m1 ,...,ms =−∞ m1 k n

k=0

e2π i

m1 k+m2 π1 (k)+···+ms πs−1 (k) n

=

k=0

s−1 ∏ 1 ∑ 2π i mν+1 πν (k) n e . n!

ν=1

πν ∈Sn

1

∑

Notice, that 1 ∑ n!

e2π i

mν+1 πν (k) n

=

πν ∈Sn

=

e2π i

mν+1 l n

n!

l=0

n−1 1∑

n

n−1 ∑

e2π i

mν+1 l n

= δn (mν+1 ) =

l=0

{

1=

πν ∈Sn ,πν (k)=l

1,

for mν+1 ≡ 0

0,

for mν+1 ̸ ≡ 0

(mod n), (mod n).

So ∞ ∑ ′

M(R[f ]) =

⃗ c(m)

m1 ,...,ms =−∞

s ∏

∞ ∑ ′

δn (mν ) =

c(nm1 , . . . , nms ).

m1 ,...,ms =−∞

ν=1

Passing to estimates modulo, we obtain ∞ ∑ ′

|M(R[f ])| ≤ ∥f (⃗x)∥Esα

1

m1 ,...,ms =−∞

(( = ∥f (⃗x)∥Esα

1+

2ζ (α )

)s

nα

(nm1 . . . nms )α

=

) s ∥f (⃗x)∥Esα ∑ ν 2ν ζ ν (α ) −1 = Cs α (ν−1) . α n

ν=1

n

Theorem is completely proved. □ 4. Conclusion Analyzing the proof of Theorem 2, one can understand that on the boundary function of the class Esα (see [17]) an estimate (11) is reached. It is the paradoxical situation when the mathematical expectation of an error for a class of nets is better than the lower bound of I. F. Sharygin for the class Esα . Naturally, this situation requires further comprehension. Using the example of this article, one can see that although Michel Desa left this World, he continues to invisibly inspire to further research in the science that he devoted all of his life.

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Acknowledgment This research was supported by Russian Foundation for Basic Research, grant 15-01-01540a. References [1] L. Carlitz, Congruences connected with three-line latin rectangles, Proc. Amer. Math. Soc. 4 (1953) 9–11. [2] M.N. Dobrovolsky, One generalization of the problem about the guests XXV scientific –pedagogical conference of mathematical departments of pedagogical institutes of Ural zone. Abstracts. Sverdlovsk, 1967, pp 11–13. [3] M.N. Dobrovolsky, Solving one of combinatorial tasks, Proc. Tula State Pedagog. Instit. 5 (1954) 190–197. [4] M.N. Dobrovolsky, On solution of one system of recurrent equations, Proc. Tula State Pedagog. Instit. 7 (1960) 220–223. [5] M.N. Dobrovolsky, The Number of Permutations of Elements of n Pairs of Two Limitations of Positions, Vol. 5, Vestn. Mosk. University, 1966, pp. 36–40. [6] M.N. Dobrovolsky, Four-line Latin rectangles Materials interuniversity Scientific Conference of Mathematical Departments of Pedagogical Institutes of the Central zone, Tula, 1968, pp. 72–75. [7] M.N. Dobrovolsky, The decision among combinatorial problems, Chebyshevskii sbornik VIII (3(23)) (2007) 31–38. [8] M.N. Dobrovolsky, On solution of one system of recurrent equations, Chebyshevskii sbornik VIII (3(23)) (2007) 39–42. [9] M.N. Dobrovolsky, Some combinatorial problems on permutations with restriction items, Chebyshevskii sbornik VIII (3(23)) (2007) 51–108. [10] M.N. Dobrovolsky, Speech on dissertations 18.01.1971, Chebyshevskii sbornik VIII (23) (2007) 47–50. [11] M.N. Dobrovolsky, Two combinatorial problems, Chebyshevskii sbornik VIII (3(23)) (2007) 14–28. [12] N.M. Dobrovolsky, I.Yu. Rebrova, A.E. Ustyan, F.V. Podsypanin, E.V. Podsypanin, Tula school of number theory (to the 105th anniversary of vladimir dmitrievich podsypanina (16.01.1910 –11.10.1968) and the 65th anniversary of the tula school of number theory), in: Algebra, Number Theory and Discrete Geometry: Contemporary Issues and Applications: Proceedings of the XIII International Conference. Supplementary volume, publishing house of Tul. GOS PED UN-TA im. L N Tolstoy, Tula, 2015, pp. 20–85. [13] S.M. Jacob, The enumeration of latin rectangle of depth three, Proc. Lond. Math. Soc. 31 (1930) 329–354. [14] K. Yamamoto, On the asymptotic number of latin rectangles, Japan. J. Math. 21 (1951) 113–119. [15] I. Kaplansky, Solution of the ‘‘probleme des menages’’, Bull. Amer. Math. Soc. 49 (1943) 784–785. [16] I. Kaplansky, J. Riordan, The probleme des menages, Scr. Math. 12 (1946) 113–124. [17] N.M. Korobov, Number-Theoretic Methods in Approximate Analysis, second ed., MTSNMO, Moscow, 2004. [18] S.M. Kerawala, The enumeration of the latin rectangle of depth three by means of a difference equation, Bull. Calcutta Math. Soc. 33 (1941) 119–127. [19] J. Riordan, Introduction to Combinatorial Analysis, Publishing house I –L, Moscow, 1963, 287 p. [20] K.A. Rybnikov, Introduction to Combinatorial Analysis, Publishing house of Moscow University, Moscow, 1972, 256 p. [21] K.A. Rybnikov, Introduction to Combinatorial Analysis/2-e Izd, Publishing house of Moscow University, Moscow, 1985, 308 p.

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