Cent. Eur. J. Math. • 12(9) • 2014 • 1390-1402 DOI: 10.2478/s11533-014-0421-2 Central European Journal of Mathematics Parity-alternating permutations and successions Research Article Augustine O. Munagi1∗ 1 School of Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, Johannesburg, South Africa Received 24 January 2013; accepted 10 September 2013 Abstract: The study of parity-alternating permutations of {1; 2; : : : ; n} is extended to permutations containing a prescribed number of parity successions – adjacent pairs of elements of the same parity. Several enumeration formulae are computed for permutations containing a given number of parity successions, in conjunction with further parity and length restrictions. The objects are classified using direct construction and elementary combinatorial techniques. Analogous results are derived for circular permutations. MSC: 05A05, 05A15 Keywords: Parity-alternating permutation • Succession block • Circular permutation © Versita Sp. z o.o. 1. Introduction p ; p ;::: n { ; ; : : : ; n} We consider the enumeration of linear permutations ( 1 2 ) of elements of the set [ ] = 1 2 with respect p ; p to pairs of entries ( i i+1) satisfying p ≡ p ; i > : i i+1 (mod 2) 0 (1) p ; p parity succession succession An ordered pair ( i i+1) satisfying (1) will also be called a , or simply a . This terminology is adapted from an analogous usage in the study of finite integer sequences (π(1); : : : ; π(N)) in which a succession refers to a pair π(i); π(i + 1) with π(i + 1) = π(i) + 1 (see for example [2, 3, 6, 8]). The concept of parity succession was first introduced in [5] in connection with certain general results in the study of alternating subsets of integers. Our definition immediately generalizes the class of parity-alternating permutations, that is, linear arrangements of distinct positive integers in which the terms assume even and odd values alternately. Since there is no pair of adjacent elements satisfying (1), a parity-alternating permutation always contains 0 successions. ∗ E-mail: [email protected] 1390 A.O. Munagi Enumeration study of permutations of [n] according to parity of individual elements is still at a budding stage. Tanimoto has applied the properties of parity-alternating permutations to the investigation of signed Eulerian numbers [7]. But we found only one relevant enumeration result, namely (2) below. In [4] the author provides permutation analogues of classical theorems on alternating subsets such as the theorems of Terquem and Skolem, with some generalizations. This paper supplies a budget of enumeration formulae for permutations classified by assorted parity criteria. Permutations are arguably the most popular objects in combinatorics, and it should not come as a surprise to find more interesting results here than those from alternating subsets. f n n The number 0( ) of parity-alternating permutations of [ ] may be stated as follows: − n n n f n 3 + ( 1) + 1 0( ) = ! ! (2) 2 2 2 We will obtain several generalizations of (2) by imposing restrictions on the number of elements of a fixed parity, the number of parity successions, and so forth (Section 2). We begin by devising a fundamental algorithm for constructing the relevant permutations (proof of Theorem 2.1). By a succession block we understand a maximal (sub-)string of elements of the same parity. A sequence of consecutive succession blocks will also be referred to as alternating. For example, the permutation (10; 1; 3; 9; 5; 12; 4; 11; 6; 2; 7; 8) has seven alternating blocks, (10); (1; 3; 9; 5); (12; 4); (11); (6; 2); (7); (8), consisting of three blocks of odd integers and four blocks of even integers, with a total of five parity successions of which two are contributed by even integers. The final section deals with the enumeration of circular permutations of [n] under similar parity restrictions (Section 3). The section opens with a brief description of these permutations. The following basic facts and notations will be used in the proofs. • The number c(n; m) of compositions (or ordered partitions) of a positive integer n into m parts is given by [1, p. 54]: c n; m n−1 ( ) = m−1 . • A permutation of [n] taken k at a time, will be referred to as a k-permutation of [n]. If a k-permutation contains r parity successions and v alternating succession blocks, then v = k − r: representing the lengths of the blocks by b ; : : : ; b k b ··· b b − ··· b − v r v 1 v , we have = 1 + + v = ( 1 1) + + ( v 1) + = + . • The number of odd and even integers in [n] will be denoted, respectively, by D(n) and E(n), that is, D(n) = b(n+1)/2c, E(n) = bn/2c. • The formula for the number p(N; k) of k-permutations of [N] has three forms, namely, N! N p(N; k) = = k! = Nk ; (N − k)! k N N N N − ··· N − k N p N; k where k denotes the falling factorial, k = ( 1) ( + 1), 0 = 1. Our choice of expression for ( ), at any time, will depend on brevity, typesetting convenience or aesthetic appeal. n • If n ≥ 0, then the binomial coefficient k will assume the value zero if k is negative or not integral. 2. Enumeration of permutations by parity successions Let f(n; k; s; r) denote the number of k-permutations of [n] containing s odd integers and r parity successions. Then f(n; 1; s; r) = 0 if r > 0 since a singleton cannot contain a succession, and f(n; 1; 0; 0) = E(n), that is, n f n; ; ; r δ ; ( 1 0 ) = r;0 2 where δij is the Kronecker delta. We also have n f n; s; s; r + 1 δ ; ( ) = r;s−1 2 s 1391 Parity-alternating permutations and successions since an s-permutation consisting of odd integers contains s − 1 successions. In every permutation enumerated by f(n; k; s; r), the number k − r of alternating blocks of even and odd integers cannot exceed 2s or 2s + 1, depending on parity. Hence s ≥ b(k − r)/2c. For an upper bound we consider an object in which the first and last entries are odd integers. If odd integers account for all the successions, then there are k − s even integers separating the blocks of odd integers. Thus in general, k − r = 2(k − s) + 1, or 2s = k + r + 1. Hence k − r k + r + 1 f(n; k; s; r) > 0 =⇒ ≤ s ≤ ; (3) 2 2 where the last quantity is independently bounded above by k and D(n). k n s r We remark that the number of -permutations of [ ] containing 0 even integers and parity successions is given by f n; k; k − s ; r ( 0 ). Theorem 2.1. Let n; k; s; r be positive integers, 2 ≤ k < n, 1 ≤ s < k, with s satisfying equation (3). Then s − 1 k − s − 1 s − 1 k − s − 1 n + 1 n f(n; k; s; r) = k − r − 1 k − r − 2 + k − r − 2 k − r − 1 : 2 s 2 k−s 2 2 2 2 Proof. We construct an enumerated object B as follows. The symbols o and e will represent an unspecified odd and s n o ; o ; : : : ; o p D n ; s even integer, respectively. Take any -permutation of [ ] consisting of odd integers ( 1 2 s), in ( ( ) ) ways, k − s n b ; b ; : : : ; bv s c s; v and a ( )-permutation of [ ] consisting of even integers. Then obtain a composition ( 1 2 1 ) of , in ( 1) ways, to represent the sequence of sizes of the blocks of odd integers in B. Let the block sizes of even integers be c ; c ; : : : ; cv B ( 1 2 0 ). So the sequence of lengths of the alternating blocks of (assuming an odd first element) has the form b ; c ; b ; c ;::: ( 1 1 2 2 ). B o ; o ; : : : ; o j To specify , we insert separators in the permutation ( 1 2 s) so that a separator immediately follows the -th j b ; b b ; b b b ;::: k − s element, where = 1 1 + 2 1 + 2 + 3 The ( )-permutation of even integers is similarly separated. Now combine the two sequences of succession blocks, alternately, in a unique manner consistent with the prescribed form of B. Let f(n; k; s; r)T denote the number of permutations counted by f(n; k; s; r) in which the first element has parity T . If T is of either parity, it follows that f n; k; s; r p D n ; s p E n ; k − s c s; v c k − s; v ; ( )T = ( ( ) ) ( ( ) ) ( 1) ( 0) that is, s − k − s − f n; k; s; r p D n ; s p E n ; k − s 1 1 : ( )T = ( ( ) ) ( ( ) ) v − v − (4) 1 1 0 1 v v B It remains to specify the lengths of the blocks of even and odd integers, 0 and 1, subject to the pattern of and the parity of k − r, the number of alternating succession blocks. Note that f n; k; s; r f n; k; s; r f n; k; s; r : ( ) = ( )odd + ( )even (5) B v v q > k − r q First assume that the first element of is odd. Then the blocks satisfy 1 = 0 = 1 0 if = 2 1, and v v q > k − r q o; : : : ; e o; : : : ; o 1 = 0 + 1 = 2 + 1 1 if = 2 2 + 1, which correspond to permutations of the form ( ) and ( ), respectively. Hence we obtain from (4), f n; k; s; r s − k − s − s − k − s − ( )odd 1 1 1 1 p D n ; s p E n ; k − s = q − q − + q q − ( ( ) ) ( ( ) ) 1 1 1 1 2 2 1 s − 1 k − s − 1 s − 1 k − s − 1 = k − r − 2 k − r − 2 + k − r − 1 k − r − 3 ; 2 2 2 2 1392 A.O.