Appendix A Useful notions and facts In this appendix we introduce most of the statistics on permutations and words ap- pearing in this book, as well as several well-known sequences of numbers and certain lattice paths. Also, we state in Table A.7 several classes of restricted permutations together with references and initial values of the corresponding sequences. A.1 Statistics on permutations and words Definition A.1.1. In a permutation π = π1π2 : : : πn, πi is a left-to-right maximum (resp., left-to-right minimum) if πi ≥ πj (resp., πi ≤ πj) for 1 ≤ j ≤ i. Also, πi is a right-to-left maximum (resp., right-to-left minimum) if πi ≥ πj (resp., πi ≤ πj) for i ≤ j ≤ n. Example A.1.2. If π = 3152647 then the left-to-right maxima are 3, 5, 6 and 7, the left-to-right minima are 3 and 1, the right-to-left maximum is 7, and the right-to-left minima are 7, 4, 2 and 1. In Table A.1 we define all permutation statistics appearing in the book. The list of statistics is essentially taken from [259]. Note that many of the statistics defined make sense for words, though not all. For example, the statistic head :i, the position of the smallest letter, is not defined for words as we may have several instances of the smallest letter. The expression π1 appearing in Table A.1 means the permutation beginning with π and ending with a letter that is larger that any letter in π. Remark A.1.3. Six statistics in Table A.1 have complex names like comp :r or last :i, which stands for respective compositions (see Definition 1.0.12 for meaning of the trivial bijections i and r). For example, to calculate the value of the statistic S. Kitaev, Patterns in Permutations and Words, Monographs in Theoretical Computer Science. 387 An EATCS Series, DOI 10.1007/978-3-642-17333-2, © Springer-Verlag Berlin Heidelberg 2011 388 Appendix A. Useful notions and facts Statistic Description asc # of ascents (letters followed by a larger letter = occurrences of 12) comp # of components (ways to factor π = στ so that each letter in non-empty σ is smaller than any letter in τ; the empty τ gives one such factorization) comp :r # of reverse components (ways to factor π = στso that each letter in non- empty σis larger than any letter in τ; the empty τgives one factorization) cyc # of cycles dasc # of double ascents (# of occurrences of the pattern 123) ddes # of double descents (# of occurrences of the pattern 321) des # of descents (letters followed by a smaller letter = occurrences of 21) exc # of excedances (positions i in π = π1π2 : : : πn such that πi > i) fp # of fixed points (positions i in π = π1π2 : : : πn such that πi = i) head the first (leftmost) letter head :i position of the smallest letter inv # of inversions (pairs i < j such that πi > πj in π = π1π2 : : : πn) last the last (rightmost) letter last :i position of the largest letter ldr length of leftmost decreasing run (largest i such that π1 > π2 > ··· > πi) lds length of the longest decreasing subsequence length # of letters lir length of leftmost increasing run (largest i such that π 1 < π2 < ··· < πi) lir :i = zeil:c, largest i such that 12 ··· i is a subsequence in π lis length of the longest increasing subsequence lmax # of left-to-right maxima lmin # of left-to-right minima maj sum of positions in which descents occur peak # of peaks (positions i in π such that πi−1 < πi > πi+1) peak :i #f i j i is to the right of both i − 1 and i + 1g slmax largest i such that π1 ≥ π1, π1 ≥ π2,..., π1 ≥ πi (# of letters to the left of second left-to-right maximum in π1) rank for π = π1π2 : : : πn, largest k such that πi > k for all i ≤ k (see [361]) rdr = lir:r, length of the rightmost decreasing run rmax # of right-to-left maxima rmin # of right-to-left minima rir = ldr:r, length of the rightmost increasing run valley # of valleys (positions i in π such that πi−1 > πi < πi+1) valley :i #f i j i is to the left of both i − 1 and i + 1g zeil length of longest subsequence n(n − 1) ··· i in π of length n (see [815]) Table A.1: Definitions of permutation statistics appearing in the book. A.2 Numbers and lattice paths of interest 389 For π = 4523176 For π = 3265714 asc(π) = 3, comp(π) = 3, des(π) = 3 asc(π) = 3, comp(π) = 2, des(π) = 3 comp :r(π) = 1, exc(π) = 3, fp(π) = 0 comp :r(π) = 1, exc(π) = 4, fp(π) = 1 head(π) = 4, head :i(π) = 5, last(π) = 6 head(π) = 3, head :i(π) = 6, last(π) = 4 last :i(π) = 6, ldr(π) = 1, lds(π) = 3 last :i(π) = 5, ldr(π) = 2, lds(π) = 3 lir(π) = 2, lir :i(π) = 1, slmax(π) = 1 lir(π) = 1, lir :i(π) = 1, slmax(π) = 2 lmax(π) = 3, lmin(π) = 3, peak(π) = 3 lmax(π) = 3, lmin(π) = 3, peak(π) = 2 peak :i(π) = 1, lis(π) = 3, rdr(π) = 2 peak :i(π) = 2, lis(π) = 3, rdr(π) = 1 rmax(π) = 2, rmin(π) = 2, cyc(π) = 2 rmax(π) = 2, rmin(π) = 2, cyc(π) = 3 rir(π) = 1, valley :i(π) = 2, zeil(π) = 2 rir(π) = 2, valley :i(π) = 2, zeil(π) = 1 dasc(π) = 0, ddes(π) = 0, maj(π) = 12 dasc(π) = 0, ddes(π) = 0, maj(π) = 9 inv(π) = 9, valley(π) = 2, rank(π) = 2 inv(π) = 10, valley(π) = 3, rank(π) = 1 Table A.2: Values of permutation statistics introduced in Table A.1 on the permu- tations 4523176 and 3265714 (length(π) = 7 in both cases). comp :r we can either follow the direct description of it in Table A.1, or we can take a permutation, apply to it the reverse operation r and calculate the value of the statistic comp on the obtained permutation. Similarly, a way to calculate last :i is to take a permutation, apply to it the inverse operation i, and to calculate the value of the statistic last on the obtained permutation. Table A.2 provides examples supporting the definitions in Table A.1. Remark A.1.4. As noted, not all permutation statistics can be translated into statistics for words. Similarly, not all word statistics can be translated into permu- tation statistics. For example, the statistic lev, which is the number of levels, is defined on words as the number of times two equal letters stay next to each other (e.g., lev(3114244333) = 4), is of interest in words, but has the trivial value zero on permutations. A.2 Numbers and lattice paths of interest In this section we collect definitions and some basic facts on selected numbers appear- ing in this book. For more information on particular objects of interest discussed below, one should consult other sources, such as standard texts in combinatorics (e.g., [743, 745]). 390 Appendix A. Useful notions and facts A.2.1 Numbers involved The Fibonacci numbers are named after Leonardo of Pisa (c. 1170 - c. 1250), who was known as Fibonacci. The Fibonacci numbers were known in ancient India, and there are many applications of these numbers. The n-th Fibonacci number is defined recursively as follows. F0 = 0, F1 = 1, and for n ≥ 2, Fn = Fn−1 + Fn−2. These numbers satisfy a countless number of identities. Letting p 1 + 5 ' = ≈ 1:6180339887 ··· 2 denote the golden ratio, the n-th Fibonacci number is given by 'n − (1 − ')n 'n − (−1=')n Fn = p = p : 5 5 p n Thus, asymptotically Fn ∼ ' = 5. The generating function for the Fibonacci numbers is given by X x F (x) = F xn = : n 1 − x − x2 n≥0 The first few Fibonacci numbers, for n = 0; 1; 2;:::, are 0; 1; 1; 2; 3; 5; 8; 13; 21; 34; 55; 89; 144;:::: The Catalan numbers are named after the Belgian mathematician Eug´ene Charles Catalan (1814{1894) and they appear in many counting problems. Catalan defined these numbers in [235] in 1838. For n ≥ 0, the n-th Catalan number is given by 1 2n (2n)! C = = : n n + 1 n (n + 1)!n! A recursive way to define the Catalan numbers is as follows: C0 = 1 and Cn+1 = Pn i=0 CiCn−i for n ≥ 0, and asymptotically, the Catalan numbers grow as 4n C ∼ p : n n3=2 π The generating function for the Catalan numbers is p X 1 − 1 − 4x C(x) = C xn = : n 2x n≥0 The first few Catalan numbers, for n = 0; 1; 2;:::, are 1; 1; 2; 5; 14; 42; 132; 429; 1430; 4862; 16796; 58786; 208012; 742900;:::: A.2 Numbers and lattice paths of interest 391 We refer the reader to www-math.mit.edu/~rstan/ec/catadd.pdf for an impres- sive collection by Stanley of different structures counted by the Catalan numbers. The Bell numbers are named after Eric Temple Bell (1883-1960). The n-th Bell number gives the number of partitions of a set with n elements. For example, there are five partitions for the 3-element set fa; b; cg: ffag; fbg; fcgg, ffag; fb; cgg, ffbg; fa; cgg, ffcg; fa; bgg, and ffa; b; cgg.