A simple bjective proof of Generalized Schur's theorem.Abstract The object of this paper is to give a simple bijective proof In combinatorics, bijective proof is a proof technique that finds a bijective function adj. 1. Involving an entire organ, as when an epileptic seizure involves all parts of the brain. 2. Not specifically adapted to a particular environment or function; not specialized. 3. version of Schur's theorem In discrete mathematics, Schur's theorem is either of two different theorems of the mathematician Issai Schur. In differential geometry, Schur's theorem is a theorem of A. Schur. stated and proved by D.M. Bressoud in the year 1980. Keywords and Phrases: Schur's theorem, Generalized version, Bijective Proof. 1. Introduction In 1980, D. M. Bressoud [4] gave a combinatorial proof The term combinatorial proof is often used in either of two senses:
Theorem 1 (Generalized Schur's theorem). Given positive integers r and m such that r < m/2, let [C.sub.r,m](n) denote de·note tr.v. de·not·ed, de·not·ing, de·notes 1. To mark; indicate: a frown that denoted increasing impatience. 2. the number of partitions of n into distinct parts [equivalent to] [+ or -] r (m) and let [D.sub.r,m](n) denote the number of partitions of n of the form [b.sub.1] + ... + [b.sub.s] such that [b.sub.i] [equivalent to] 0, [+ or -]r(m), [b.sub.i] - [b.sub.i+1] [greater than or equal to] m, and [b.sub.i] - [b.sub.i+1] [greater than or equal to] 2m when [b.sub.i] [equivalent to] [b.sub.i+1] [equivalent to] 0(m). Then [C.sub.r,m](n) = [D.sub.r,m](n) for all n. In the year 2003, Padmavathamma and M. Ruby ruby, precious stone, the transparent red variety of corundum, found chiefly in Myanmar, Thailand, and Sri Lanka and classified among the most valuable of gems. The Myanmarese stones are blood red, the most valued tint being the "pigeon's blood. Salestina [5] gave a different combinatorial proof of the above theorem for the case when m = 4 and r = 1. The object of this paper is to give a simple bijective proof of Theorem 1. 2. Proof We construct a mapping from the partitions enumerated This term is often used in law as equivalent to mentioned specifically, designated, or expressly named or granted; as in speaking of enumerated governmental powers, items of property, or articles in a tariff schedule. by [C.sub.r,m](n) to those enumerated by [D.sub.r,m](n). Let [pi] = [b.sub.1] + [b.sub.2] + ... + [b.sub.s] denote a partition A reserved part of disk or memory that is set aside for some purpose. On a PC, new hard disks must be partitioned before they can be formatted for the operating system, and the Fdisk utility is used for this task. enumerated by [C.sub.r,m](n). If every pair of [b.sub.i] and [b.sub.i+1] satisfies [b.sub.i] - [b.sub.i+1] [greater than or equal to] m, then [pi] is a partition enumerated by [D.sub.r,m] also. We adopt the following procedure to map the rest of partition of [C.sub.r,m](n) into [D.sub.r,m](n). Step C[D.sub.1] : List the parts of [pi] in a column in decreasing order. Let [[pi].sup.1] denote this partition. Step C[D.sub.2] : From the top look for the first i say [alpha] for which [b.sub.[alpha]] - [b.sub.[alpha]+1] < m. The only two possibilities are: (i) [b.sub.[alpha]] = m(k + 1) - r and [b.sub.[alpha]+1] = mk + r or (ii) [b.sub.[alpha]] = mk + r and [b.sub.[alpha]+1] = mk - r In both cases we replace the two consecutive parts [b.sub.[alpha]] and [b.sub.[alpha]+1] with just one part ([b.sub.i.sub.1] + [b.sub.[i.sub.1]+1]). We note that the sum will always be [equivalent to] 0(m). In the first case ([b.sub.[alpha]] + [b.sub.[alpha]+1]) = m(2k + 1) while in the second case ([b.sub.[alpha]] + [b.sub.[alphal]+1]) = m(2k). Eg: Let m = 5 and r = 1 [MATHEMATICAL EXPRESSION A group of characters or symbols representing a quantity or an operation. See arithmetic expression. NOT REPRODUCIBLE re·pro·duce v. re·pro·duced, re·pro·duc·ing, re·pro·duc·es v.tr. 1. To produce a counterpart, image, or copy of. 2. Biology To generate (offspring) by sexual or asexual means. IN ASCII ASCII or American Standard Code for Information Interchange, a set of codes used to represent letters, numbers, a few symbols, and control characters. Originally designed for teletype operations, it has found wide application in computers. ] Let [[pi].sup.2] denote the resulting partition. We now get two possibilities. Case 1: [b.sub.[alpha]-1] - ([b.sub.[alpha]] + [b.sub.[alpha]+1]) < m. Case 2: [b.sub.[alpha]-1] - ([b.sub.[alpha]] + [b.sub.[alpha]+1]) > m We note that the possibility that [b.sub.[alpha]-1] - ([b.sub.[alpha]] + [b.sub.[alpha]+1]) = m will not arise since [b.sub.[alpha]-1] [not equivalent to] 0(m) and ([b.sub.[alpha]] + [b.sub.[alpha]+1]) [equivalent to] 0(m). In case 1, we replace the pair [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Eg: Let m = 8 and r = 3 [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Once again we get two possibilities for case 1. [b.sub.[alpha]-2] - ([b.sub.[alpha]] + [b.sub.[alpha]+1] + m) < m and [b.sub.[alpha]-2] - ([b.sub.[alpha]] + [b.sub.[alpha]+1] + m) > m. As before, in the first case we apply the procedure explained in case 1. This procedure is continued until we meet the second case or we find ([b.sub.[alpha]] + [b.sub.[alpla]+1] + tm) on the top. In case 2 from the top we look for the next i say [beta] for which [b.sub.[beta]] - [b.sub.[beta]+1] < m and we follow the same procedure explained in Step C[D.sub.2] until we meet the second case for [beta]. The requirement of the minimal difference between multiples of m is clearly satisfied in our mapping for the following reason. Let [pi] = (..., a, b,..., c, d,...) where (a, b) and (c, d) are two consecutive pairs who need to be treated by Step C[D.sub.2]. Clearly, a + b [greater than or equal to] c + d + (t + 2)m where t counts the number of parts between b and c. The procedure still needs to lift parts (a, b) and (c, d) up if necessary. For every lifting, each part is increased by m; but there is no way to lift the part caused by (c, d) above the one caused by (a, b). Therefore, the final two parts caused by (a, b) and (c, d) must have minimal difference 2m. Following the procedure explained in Step C[D.sub.2] (in a finite finite - compact number of steps) we arrive at a stage where difference condition is satisfied for all the parts of [pi]. We associate this resulting partition [[pi].sup.4] which is enumerated by [D.sub.r,m](n) to [pi]. We illustrate our procedure by an example by taking m = 5 and r = 2. Let [pi] = 47 + 42 + 38 + 37 + 28 + 27 + 23 + 18 + 13 + 12 + 3 + 2 be a partition enumerated by [C.sub.2,5] (290). [C.sub.r,m](n) [right arrow] [D.sub.r,m](n) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] The last partition is the associated partition of [pi] enumerated by [D.sub.2,5] (290). We now give the reverse mapping from [D.sub.r,m](n) to [C.sub.r,m](n). Let [psi PSI - Portable Scheme Interpreter ] be a partition enumerated by [D.sub.r,m](n). If no part is a multiple of m, then it is a partition enumerated by [C.sub.r,m](n) also. We adopt the following procedure to map the rest of partition of [D.sub.r,m](n) into [C.sub.r,m](n). Step D[C.sub.1] : Let the parts of [psi] be arranged in a column in decreasing order. Let [[psi].sup.1] denote this partition. Step D[C.sub.2] : From the bottom look for the first multiple of m say x. We split x into ([alpha], [beta]) tentatively as below: TABLE x = m * (2k) [right arrow] (mk + r, mk - r). x = m * (2k + 1) [right arrow] (m(k + 1) - r, mk + r). Suppose y lies below x. If y < [beta] then the tentative tentative, adj not final or definite, such as an experimental or clinical finding that has not been validated. splitting is just what we want; otherwise, we replace [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Now split x - m into ([alpha], [beta]) tentatively as before, and then apply the same procedure on x - m and the part below it. This process is continued till the end. Let the resulting partition be [[psi].sup.2]. Eg1: Let m = 8 and r = 3 [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Eg2: Let m = 8 and r = 1 [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Step D[C.sub.2] will not create multiples of m. This is obvious if y < [beta]. When y [greater than or equal to] [beta], the step involves only addition or subtraction subtraction, fundamental operation of arithmetic; the inverse of addition. If a and b are real numbers (see number), then the number a−b is that number (called the difference) which when added to b (the subtractor) equals of m which does not change the congurecy of x or y (mod m). From the bottom look for the next multiple of m say [x.sup.1] and follow the same procedure explained in Step D[C.sub.2] to split [x.sup.1]. We apply Step D[C.sub.2] until all the multiples of [psi] are split into parts [equivalent to] [+ or -]r(m). The resulting partition will be a partition enumerated by [C.sub.r,m](n). We also claim: Let [pi] = (..., x,..., y,...) where x and y are two consecutive multiples of m. Clearly, x [greater than or equal to] y + (t + 2)m where t counts the number of parts between x and y. During the procedure x and y would be moved downward with m subtracted each time. However, the splitting caused by x will never go under nor between the ones caused by y. This is obvious because if the resulting parts obtained are x' and y' then x' will be [greater than or equal to] y' + 2m always. And [beta] part of x' will be > [alpha] part of y'. We now illustrate the reverse map by taking the same partition, [psi] = 85 + 65 + 45 + 32 + 27 + 18 + 13 + 5 where m = 5 and r = 2 obtained from [psi] = 47 + 42 + 38 + 37 + 28 + 27 + 23 + 18 + 13 + 12 + 3 + 2 [D.sub.r,m](n) [right arrow] [C.sub.r,m](n) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] The above two mappings [C.sub.r,m](n) [right arrow] [D.sub.r,m](n) and [D.sub.r,m](n) [right arrow] [C.sub.r,m](n) are inverse (mathematics) inverse - Given a function, f : D -> C, a function g : C -> D is called a left inverse for f if for all d in D, g (f d) = d and a right inverse if, for all c in C, f (g c) = c and an inverse if both conditions hold. to each other follows from the reasons mentioned below. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], since x [greater than or equal to] mk + r + m [left and right arrow] x - m [greater than or equal to] mk + r which is [beta] part of m(2k + 1). [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] since x [greater than or equal to] m(k + 1) - r + m [left and right arrow] x - m [greater than or equal to] m(k + 1) - r which is [beta] part of m(2k + 2). References [1] A. J. Yee, A combinatorial proof of Andrews' partition functions
n. Something printed and often distributed in partial or preliminary form in advance of official publication: a preprint of a scientific article. tr.v. ). [2] K. Alladi and B. Gordon, Generalizations of Schur's partition theorem, Manus MANUS. Anciently signified the person taking an oath as a compurgator. The use of this word probably came from the party laying his hand on the New Testament. Manus signifies, among the civilians, power, and is frequently used as synonymous with potestas. Lec. El. Dr. Rom. Sec. 94. . Math. 79 (1993), 113-126. [3] C. Bessenrodtoud, A combinatorial proof of a refinemetn of the Andrews-Olsson partition identity, European J. Combin. 12 (1991), 271-276. [4] D. M. Bressoud, A combinatorial proof of Schur's 1926 partition theorem, Proc. Amer. Math. Soc., Vol.79, 2 (1980), 338-340. [5] Padmavathamma and M. Ruby Salestina, A simple proof of Schur's 1926 theorem on partitions, Applied Science Periodical periodical, a publication that is issued regularly. It is distinguished from the newspaper in format in that its pages are smaller and are usually bound, and it is published at weekly, monthly, quarterly, or other intervals, rather than daily. , SIWAN, Bihar Coordinates: Siwan is the district headquarters of the Siwan district in the Indian state of Bihar. Geography Siwan (the old name Alipur)is located at . , Vol V, 3 (August 2003), 135-138. Padmavathamma, ([dagger]) R. Raghavendra, ([double dagger double dagger n. A reference mark (  ) used in printing and writing. Also called diesis.Noun 1. ]) and B. M. Chandrashekara ([section]) Department of Studies in Mathematics, University of Mysore University of Mysore is a public university in India. It has its main campus in the city of Mysore and extension campuses in the neighbouring districts of Hassan, and Mandya. , Manasagangotri, Mysore 570 006, Karnataka, India Received April 22, 2005, Accepted July 26, 2006. *2000 Mathematics Subject Classification. Primary 11P81, 11P82, 11P83; Secondary 05A15, 05A17, 05A19 ([dagger]) E-mail: vathamma@yahoo.com ([double dagger]) E-mail: maths@yahoo.co.in ([section]) E-mail: alur@yahoo.com |
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