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The Statistic Pinv for Number System∗

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The Statistic Pinv for Number System∗ Applied Mathematics E-Notes, 20(2020), 230-235 c ISSN 1607-2510 Available free at mirror sites of http://www.math.nthu.edu.tw/ amen/ The Statistic Pinv For Number System Fanomezantsoa Patrick Rabarisony, Hery Randriamaroz Received 19 April 2019 Abstract The number of inversions is a statistic on permutation groups measuring the degree to which the entries of a permutation are out of order. We provide a generalization of that statistic by introducing the statistic number of pseudoinversions on the colored permutation groups. The main motivation for studying that statistic is the possibility to use it to define a number system and a numeral system on the colored permutation groups. By means of the statistic number of i-pseudoinversions, we construct our number system, and a bijection between the set of positive integers and the colored permutation groups. 1 Introduction A statistic over a group is a function from that group to the set of nonnegative integers. One of the most studied statistics is the number of inversions, defined by inv := # (i, j) [n]2 i < j, (i) > (j) , 2 j on the symmetric group Sn over [n]. A well-known result is the equidistribution of inv with the statistic major index proved by Foata [3]. Besides, the Lehmer code is based on that statistic. It is a particular way to encode each permutation of n numbers, and an instance of a scheme for numbering permutations. Remark that Vajnovszki provided several permutation codes directly related to the Lehmer code [5]. In this article, we give a generalization of the statistic inv on the colored permutation group in order to create a more general code. The colored permutation group of m colors and n elements is the wreath product th Um Sn of the group Um of all m roots of unity by the symmetric group Sn on [n]. We represent a colored o permutation Um Sn by 2 o k 1 2 . n 2i j = with Sn and = e m . (1) (2) . (n) 2 kj k1 k2 kn For i, j Z such that i < j, let [i, j] := i, i + 1, . , j . 2 f g Definition 1 The number of i-pseudoinversions of Um Sn is 2 o pinv := ki(n i + 1) + j [i + 1, n] (i) > (j) . i 2 And the number of pseudoinversions of Um Sn is 2 o n pinv := pinvi . i=1 X 1 2 3 4 Example 1 Consider the element = U5 S4. We have pinv1 = 1, pinv2 = 3, 2 1 4 3 2 o 1 4 pinv3 = 9, pinv4 = 0, and pinv = 13. The interest for investigating statistics on Um Sn recently arised. Bagno et al., for example, introduced the statistics (c, d)-descents and computed their distributionso [1, Proposition 1.1.]. We construct a number system by means of the cardinality Um Sn , and the statistic pinvi. j o j Mathematics Subject Classifications: 05A19. yDépartement de Mathématiques et Informatique, Université d’Antananarivo, 101 Antananarivo, Madagascar zDépartement de Mathématiques et Informatique, Université d’Antananarivo, 101 Antananarivo, Madagascar 230 F. P. Rabarison and H. Randriamaro 231 Definition 2 Let A = (Ai)i N, a = (ai)i N be two sequences of positive integers. The pair (A, a) is a number 2 2 k system if, for every integer n N, there exists k N such that n = iAi with i [0, ai], and this 2 2 2 i=0 X representation in terms of Ai’s and i’s is unique. Using formal power series, Cantor provided a condition for a pair of positive integer sequences to be a number system [2, §.2.]. We provide a more suitable condition for this work. Proposition 1 Let A = (Ai)i N and a = (ai)i N be two sequences of strictly positive integers. The pair of sequences (A, a) is a number system2 if and only2 if k 1 A0 = 1 and Ak = (1 + ai) for each k N. 2 i=0 Y We prove Proposition1 in Section2. Corollary 1 Let m N. The pair of sequences (G, g) = (Gi)i N, (gi)i N defined by 2 2 2 i Gi = m i! and gi = m(i + 1) 1 is a number system. Proof. We obtain the equality in Proposition1 for number system from k 1 k 1 k (1 + gi) = m(i + 1) = m k! = Gk. i=0 i=0 Y Y We particularly use the number system (G, g) in this article. The number Gi is the cardinality of Um Si, o and gi is the maximal value of pinv1 relating to Um Si+1. The factorial number system introduced by Laisant [4] corresponds to the case m = 1 of (G, g). Weo also construct a numeral system by means of the groups Um Sn and the statistic pinvi. o Definition 3 A numeral system is a notation for representing integers. There exist several types of numeral systems depending on the historical context and the geographical location. We develop a numeral system based on the colored permutation groups. Write k k 1 1 0 k ··· for the integer Gi, and G, g k for the integer set i=0 i h i P G, g k := k k 1 1 0 i [0, gi] . h i ··· 2 Our numeral system stems from the following bijection. Theorem 1 Let n 1. The following map is bijective ≥ Um Sn G, g n 1 g : . o !pinv () pinv h (i ) pinv () 7! 1 2 ··· n We prove Theorem1 in Section3. The Lehmer code is built from the case m = 1. 1 2 3 4 Example 2 From Example1, we have = = 1 3 9 0 which, is equal to 945 in 2 11 44 3 decimal system. 232 The Statistic pinv Furthermore, we obtain the generating function of the statistic number of pseudoinversions. Denote the i 1 q-analog of the number i by [i]q := 1 + q + + q . ··· Corollary 2 The generating function of the statistic pinv on Um Sn is o n pinv q = [mi]q. Um Sn i=1 2Xo Y Proof. The bijection of Theorem1 implies that every element of Um Sn has a unique representation in o G, g n 1. Then, we have h i n pinv q = [max pinvn + 1]q ... [max pinv2 + 1]q [max pinv1 + 1]q = [mi]q. Um Sn i=1 2Xo Y 2 Proof of Proposition1 We provide a condition for a pair of integer sequences (A, a) to be a number system. Lemma 1 Take a number system (A, a), and let n = k 1 0 be a nonnegative integer. Then, ··· kAk n < ( k + 1)Ak. ≤ k 1 Proof. It is clear that kAk n. Suppose that n ( k + 1)Ak which means i Ai Ak. Then, there ≤ ≥ ≥ i=0 k timesX exist i’s, i [0, k 1], such that 0 i i and k 1 1 0 = 1 0 0 0. That contradicts the uniqueness of2 the representation. ≤ ≤ ··· ··· z }| { Lemma 2 Let A = (Ai)i N, a = (ai)i N be two sequences of positive integers. Then, the pair (A, a) is a number system if and only2 if 2 k 1 A0 = 1 and Ak = aiAi + 1 for each k N. 2 i=0 X Proof. Suppose that (A, a) is a number system. It is obvious that we must have A0 = 1. Let n = ak a1a0. k ··· From the second inequality of Lemma1, we get aiAi = n < (ak + 1)Ak. Then, i=0 X k k 1 aiAi + 1 (ak + 1)Ak i.e. aiAi + 1 Ak. ≤ ≤ i=0 i=0 X X k 1 k 1 As aiAi + 1 / A, a k 1, the only possibility is aiAi + 1 = Ak. 2 h i i=0 i=0 X X k 1 Now, suppose that A0 = 1 and Ak = aiAi + 1 for each k N. Let n N, and assume that every 2 2 i=0 X k integer m n 1 has a unique representation k k 1 1 0 with i [0, ai]. If n 1 = iAi, ≤ ··· 2 i=0 X F. P. Rabarison and H. Randriamaro 233 let j = max l [0, k] i [0, l]: i = ai . Then, 2 8 2 k n = Aj+1 + iAi. i=j+1 X k k0 About the uniqueness, suppose that n has two different representations iAi and i0 Ai: i=0 i=0 X X k0 k0 1 k If k0 > k, then 0 Ai 0 Ak = ( 0 1)Ak + aiAi + 1 > iAi. • i ≥ k0 0 k0 0 i=0 i=0 i=0 X X X Otherwise, let l = max i [0, k] i = 0 , and assume that 0 > l. We have • 2 6 i l k l 1 k i0 Ai = i0 Ai + l0Al + i0 Ai i=0 i=0 i=l+1 X X X k 0 Ai + 0Al ≥ i l i=l+1 X k l 1 = iAi + ( 0 1)Al + aiAi + 1 l i=l+1 i=0 X X k > iAi. i=0 X As that result is absurd, it is therefore impossible to have two different representions of n. We can now proceed to the proof of Proposition1: From Lemma2, we deduce that (A, a) is a number system if and only if A0 = 1 and k 1 k 2 Ak = aiAi + 1 = ak 1Ak 1 + aiAi + 1 = ak 1Ak 1 + Ak 1 i=0 i=0 X X k 1 =(ak 1 + 1)Ak 1 = (ak 1 + 1)(ak 2 + 1) ... (a0 + 1) = (1 + ai). i=0 Y 3 Proof of Theorem1 We finally prove that there is a one-to-one correspondence betweenthe permutations 1 2 .
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