A permutation code preserving a double Eulerian bistatistic

A permutation code preserving a double Eulerian bistatistic Jean-Luc Baril and Vincent Vajnovszki LEI, Université de Bourgogne Franche-Comté BP 4787, 78 Dijon Cedex, France {barjl}{vvajnov}@u-bourgogne.fr February, 7 Abstract In 977 Foata proved bijectively, among other things, that the joint distribution of ascent and distinct nonzero value numbers on the set of subexcedant sequences is the same as that of descent and inverse descent numbers on the set of permutations, and the generating function of the corresponding bistatistics is the double Eulerian polynomial. In Foata s result was rediscovered by Visontai as a conjecture, and then reproved by Aas in 4. In this paper we define a permutation code (that is, a bijection between permutations and subexcedant sequences) and show the more general result that two 5-tuples of setvalued statistics on the set of permutations and on the set of subexcedant sequences, respectively, are equidistributed. In particular, these results give another bijective proof of Foata s result. Introduction In enumerative combinatorics it is a classical result that the descent number des and the inverse descent number ides (defined as idesπ = desπ ) on permutations are Eulerian statistics, and their distributions on the set S n of length-n permutations are given by the nth Eulerian polynomial A n, that is A n (u) = u desπ+ = u idesπ+, π S n π S n and the joint distribution of des and ides is given by the nth double Eulerian polynomial, A n (u,v) = π S n u desπ+ v idesπ+, see for instance [, 8]. An alternative way to represent a permutation is its Lehmer code [5], which is a subexcedant sequence. The ascent number asc on the set S n of subexcedant sequences is still an Eulerian statistic (see for example [9]), and in [7] the statistic that counts the number of distinct nonzero symbols in a subexcedant sequence (that following [] we denote by row) is proved to be still Eulerian; this result is credited to Dumont by the authors of [7]. In terms of generating functions we have A n (u) = u ascs+ = u rows+. s S n s S n

In 977 Foata [4] proved, among other things, that the joint distribution of des and ides on the set of permutations is the same as that of asc and row on the set of subexcedant sequences, that is A n (u,v) = u desπ+ v idesπ+ = u ascs+ v rows+. π S n s S n In the present paper we define a bijection between permutations and subexcedant sequences (i.e., a permutation code) and show that the tuple of set-valued statistics (Des, Ides, Lrmax, Lrmin, Rlmax) on the set of permutations, and (Asc, Row, Posz, Max, Rlmax) on the set of subexcedant sequences have the same distribution (each of the occurring statistics is defined below). In particular, our bijection provides set-valued partners for Asc (answering to a question in []) and gives a alternative proof of Foata s result. Notation and definitions A length-n word w over the alphabet A is a sequence w w...w n of symbols in A, and we will consider only finite alphabets A N. Statistics A statistic on a set X of words is simply a function from X to N; a set-valued statistic is a function from X to N ; and a multistatistic is a tuple of statistics. Let w = w w...w n be a length-n word. A descent in w is a position i in w, i < n, with w i > w i+, and the descent set of w is Desw = {i : i < n with w i > w i+ }. A left-to-right maximum in w is a position i in w, i n, with w j < w i for all j < i, and the set of left-to-right maxima is Lrmaxw = {i : i n with w j < w i for all j < i}. Clearly Lrmaxw, and Des and Lrmax are classical examples of set-valued statistics on words. We define similarly the sets Ascw of ascents, Lrminw of left-to-right minima, Rlmaxw of right-to-left maxima and Rlmin w of right-to-left minima in w. To each set-valued statistic St corresponds an (integer-valued) statistic st defined as st w = cardstw, for example desw and lrmaxw counts, respectively, the number of descents and the number of left-to-right maxima in w. Let X and X be two sets of words, and st and st be two statistics defined on X and X, respectively. We say that st on X has the same distribution as st on X (or equivalently, st and st are equidistributed) if, for any integer u, card{w X : stw = u} = card{w X : st w = u}, and the multistatistic (st,st,...,st p ) defined on X has the same distribution as the multistatistic (st,st,...,st p) defined on X (or the two multistatistics are equidistributed) if, for any integer p-tuple u = (u,u,...,u p ), card{w X : (st,st,...,st p )w = u} = card{w X : (st,st,...,st p)w = u}.

The notion of equidistribution of (multi)statistics can naturally be extended to set-valued (multi)statistics. Permutations, subexcedant sequences and codes This paper deals with two particular classes of words: permutations and subexcedant sequences. A permutation is a length-n word over {,,...,n} with distinct symbols. Alternatively, a permutation is an element of the symmetric group on {,,...,n} written in one line notation, and S n denotes the set of length-n permutations. If two permutations π = π π...π n and σ = σ σ...σ n are such that σ π σ π σ πn = n (i.e., the identity in S n ), then σ is the inverse of π, which is denoted by π. A length-n subexcedant sequence is a word s = s s...s n over {,,...,n } with s i i for i n, and S n denotes the set of length-n subexcedant sequences; and we have S n = {} {,} {,,...,n }. Some statistics are consistently defined only on particular classes of words, e.g. permutations or subexcedant sequences. For a permutation π S n, an inverse descent (ides for short) in π is a position i for which π i + appears to the left of π i in π. Equivalently, i is an ides in π if π i is a descent in π. The ides set is defined as Idesπ = {i : < i n with π i + appears in π to the left of π i }, and idesπ = desπ, but in general Idesπ is not equal to Desπ. Let s = s s...s n be a subexcedant sequence in S n. The Posz statistic gives the positions of s in s, Poszs = {i : i n,s i = }, and obviously Poszs. The Max statistic is defined as Maxs = {i : i n,s i = i }, and as above, Maxs. A last-value position in s is a position i in s such that s i and s i does not occur in the suffix s i+ s i+...s n of s. The last-value position set, denoted by Row, is defined as Rows = {i : s i and s i does not occur in the suffix s i+ s i+...s n }. Clearly / Rows, and rows = cardrows counts the number of distinct nonzero symbols in s. Example. If π = 65874 S 8 and s = 6 S 8, then Desπ = Ascs = {,4,5,6}, Idesπ = Rows = {,5,7,8}, Lrmaxπ = Poszs = {,4}, Lrminπ = Maxs = {,,7}, known in literature also as inversion sequence, inversion table or subexceedant function

Rlmaxπ = Rlmins = {4,5,8}. Aninversioninapermutationπ = π π...π n S n isapair(i,j)withi < j andπ i > π j. The set S n is in bijection with S n, and any such bijection is called permutation code. The Lehmer code L defined in [5] is a classical example of permutation code; it maps each permutation π = π π...π n to a subexcedant sequence s s...s n where, for all j, j n, s j is the number of inversions (i,j) in π (or equivalently, the number of entries in π larger than π j and on its left). For example L(65874) = 464. See also [] for a family of permutation codes in the context of Mahonian statistics on permutations. In [] is showed that dmc statistic which counts the number of distinct nonzero symbols in the Lehmer code of a permutation π (the statistic π rowl(π) with the above notations) is Eulerian, and so has the same distribution as des, asc or ides on S n. See also [] where Dumont s statistic dmc is extended to words. Although the following properties are folklore, they are easy to check. Property. If π S n and L(π) S n is its Lehmer code, then Desπ = AscL(π), Lrmaxπ = PoszL(π), Lrminπ = MaxL(π), and Rlmaxπ = RlminL(π). The permutation code b We define a mapping b: S n S n and Theorem shows that b is a bijection, that is, a permutation code, and it is the main tool in proving that (Des,Ides,Lrmax,Lrmin,Rlmax) on S n has the same distribution as (Asc,Row,Posz,Max,Rlmax) on S n (see Theorem ). A position i in π = π π...π n S n, i n, can satisfy the following properties: P: π i + occurs in π at the right of π i, P: π i occurs in π at the right of π i, where π is the permutation of {,,...,n} obtained by adding a at the end of π. And to each position i in π we associate an integer λ i (π) {,,,} according to i satisfies both, one, or none of these properties:, if i satisfies both P and P,, if i satisfies P but not P, λ i (π) =, if i satisfies P but not P,, if i satisfies neither P nor P, and we denote it simply by λ i when there is no ambiguity. Alternatively, using the Iverson bracket notation ([P] = iff the statement P is true), we have the more concise expression: λ i = [π i = n or π i + occurs at the left of π i ] + [π i occurs at the left of π i ]. For example, for any π S n we have λ = except λ = if π = n; and when n >, then λ n = except λ n = if π n =. Each λ i is uniquely determined by π, for instance if π = 65874, then λ,λ,...,λ 8 =,,,,,,,, see Figure. An interval I = [a,b], a b, is the set of integers {x : a x b}; and a labeled interval is a pair (I,l) where I is an interval and l and integer. In order to give the construction of the mapping b we define below the slices of a permutation, and some of their properties are given in Remark. 4

Definition. For a permutation π = π π...π n S n and an i, i < n, the ith slice of π is the sequence of labeled intervals U i (π) = (I,l ),(I,l ),...,(I k,l k ), defined by following process (see Figure ). U (π) = ([,n],). For i, let U i (π) = (I,l ),(I,l ),...,(I k,l k ) be the (i )th slice of π and v, v k, be the integer such that π i I v. The ith slice U i (π) of π is defined according to λ i : If λ i = (or equivalently, mini v < π i < maxi v ), then U i (π) = (I,l ),...,(I v,l v ),(H,l v ),(J,l v+ ),(I v+,l v+ ),...,(I k,l k ),(I k,l k +), where H = [π i +,maxi v ] and J = [mini v,π i ]; If λ i = (or equivalently, mini v < maxi v = π i ), then U i (π) = (I,l ),...,(I v,l v ),(J,l v+ ),(I v+,l v+ ),...,(I k,l k ),(I k,l k +), where J = [mini v,π i ]; If λ i = (or equivalently, mini v = π i < maxi v ), then U i (π) = (I,l ),...,(I v,l v ),(J,l v ),(I v+,l v+ ),...,(I k,l k ),(I k,l k +), where J = [π i +,maxi v ]; If λ i = (or equivalently, mini v = maxi v = π i ), then U i (π) = (I,l ),...,(I v,l v ),(I v+,l v+ ),...,(I k,l k ),(I k,l k +). Example. For the permutation π = 65874 in Figure, λ (π),λ (π),...,λ 8 (π) =,,,,,,,, and the process described in Definition gives the slices below. U (π) = ([,8],); U (π) = ([7,8],),([,5],); U (π) = ([7,8],),([,5],),([,],); U (π) = ([7,8],),([,4],),([,],); U 4 (π) = ([7,7],),([,4],),([,],4); U 5 (π) = ([,4],),([,],5); U 6 (π) = ([4,4],),([,],6); U 7 (π) = ([4,4],),([,],7). 8 7 6 5 4 U(π) U(π) U(π) U(π) U4(π) U5(π) U6(π) U7(π) 4 5 6 7 Remark. Let U i (π) = (I,l ),(I,l ),...,(I k,l k ) be the ith slice of π, i < n. Then, the following properties can be easily checked: the intervals I,I,...,I k are in decreasing order, that is maxi j+ < mini j for any j, j < k; the sequence l,l,...,l k is increasing, that is l j < l j+ for any j, j < k; 5

λ i = λ i = λ i = λ i = U i (π) U i (π) U i (π) U i (π) U i (π) U i (π) Iv l v l v l v l v l v l v l v Iv l v l v+ l v l v l v+ Iv+ l v+ l v+ l v+ l v+ l v+ l v+ Ik l k l k l k l k l k l k Ik l k lk + lk + lk + and l k lk + Figure : The four cases in Definition. 8 7 6 5 4 4 5 6 4 5 6 7 8 Figure : The permutation π = 65874 with b(π) = 6 and λ (π),λ (π),...,λ 8 (π) =,,,,,,,. {l,l,...,l k } [,i], and l k = i; I k ; k j= I j = {π i+,π i+,...,π n } {}; the (i+)th entry of the Lehmer code of π is given by the number of entries π j > π i+, with j < i+, that is the cardinality of [π i+,n]\ k j= I j. A byproduct of Definition is the construction of b: S n S n defined below. Definition. Let π = π π...π n S n. For each i, i n, we define b i = l v, where v is such that (I v,l v ) is a labeled interval in the (i )th slice of π with π i I v, and we denote by b(π) the sequence b b...b n. 6

From Remark it follows that b(π) is a subexcedant sequence, see for instance Example and Figure. Proposition. Let π = π π...π n S n, b(π) = b b...b n, and let an i, i n.. i is a descent in π iff i is an ascent in b(π);. i is an ides in π iff b i does not occur in b i+ b i+...b n ;. i is a left-to-right maximum in π iff b i = ; 4. i is a left-to-right minimum in π iff b i = i ; 5. i is a right-to-left maximum in π iff i is right-to-left minimum in b. Proof. Points and obviously follow from the definition of b. Point. Let j be such that π j = n; b i = iff i j and π i lies in the first interval of the (i )th slice of π, which in turn is equivalent to i is a left-to-right maximum in π. Point 4. Similarly, let j be such that π j = ; b i = i iff i j and π i lies in the last interval of the (i )th slice of π, which in turn is equivalent to i is a left-to-right minimum in π. Point 5. By the construction of b, i is a right-to-left maximum in π iff π i is the largest element of the first interval of the (i )th slice of π, which in turn is equivalent to b i is smaller than any of b i+,b i+,...,b n. See for instance Example, where s = b(π). For a length-n subexcedant sequence b = b b...b n let consider the following properties that a position i, i n, can satisfy: R: b i occurs in the suffix b i+ b i+...b n of b, R: i occurs in b. The next proposition shows that each λ i (π) can be obtained solely from b(π). Proposition. Let π S n and b(π) = b b...b n. Then for any i, i n, we have:, if i satisfies both R and R,, if i satisfies R but not R, λ i (π) =, if i satisfies R but not R,, if i satisfies neither R nor R. Proof. By the construction given in Definition for b(π) from the slices of π, it follows that the position i in π satisfies property P (resp. P) if and only if the position i in b(π) satisfies property R (resp. R), and the statement holds. Proposition. Let π,σ S n with b(π) = b(σ). Then. λ i (π) = λ i (σ) for any i, i n.. If (I,l ),(I,l ),...,(I k,l k ) is the ith slice of π, and (J,m ),(J,m ),...,(J p,m p ) that of σ, for some i, i < n, then k = p and l j = m j, for j k. 7

Proof. The first point is a consequence of Proposition. The second point follows by the next considerations. The ith, slice of π, i < n has the same number of intervals as its (i )th slice, except in two cases: λ i (π) = (when an interval is split into two intervals); and when λ i (π) = (when a one-element interval is removed). The result follows by considering the first point and by induction on i. The sequence b(π) = b b...b n S n was defined by means of the slices of π, but in proving the bijectivity of b we need rather the complement of these slices. Let π S n and U i (π) = (I,l ),(I,l ),...,(I k,l k ) be the ith sliceof π for an i, i < n. The ith profile of π isthesequencex,x,...,x p ofdecreasingnonemptymaximalintervals(thatis, maxx j+ < minx j, and none of them has the form X j X j+ ) with p j= X j = {,,...,n}\ k j= I j. And clearly, p j= X j is the set of entries in π to the left of π i+, and p j= cardx j = i. Example. The vertical grey regions on the right side of Example correspond to the profiles of π = 65874 in Figure. These profiles are: [6,6]; [6,6],[,]; [5,6],[,]; [8,8],[5,6],[,]; [5,8],[,]; [5,8],[,]; and [5,8],[,]. In the proof of Theorem we need the next result. Proposition 4. Let π,σ S n with b(π) = b(σ), and let an i, i < n. If X,X,...,X p and Y,Y,...,Y m are the ith profiles of π and of σ, then n X if and only if n Y, p = m, and cardx j = cardy j for any j, j p. Proof. Itiseasytoseethatn X iffdoesnotappearinb i+ (π)...b n (π) = b i+ (σ)...b n (σ), that is, iff n Y. And if i =, then the first profile of π and of σ are one-element intervals, and the statement holds. From the first point of Proposition we have λ i (π) = λ i (σ). Let suppose that the statement is true for i, and we will prove it for i. In passing from the (i )th profiles of π and of σ to their ith profiles, the following cases can occur (we refer the reader to Definition and Figure ). If λ i (π) = λ i (σ) =, or λ i (π) = λ i (σ) = and b i (π) = b i (σ) =, then a new oneelement interval is added to the ith profile of π and of σ. Moreover, since b(π) = b(σ), by the second point of Proposition, it follows that these intervals are both, for some k, the kth intervals in the ith profile of π and σ. If λ i (π) = λ i (σ) = and b i (π) = b i (σ), then for some k, a new element is added to the kth interval of both ith profiles of π and σ; this element is the smallest one in the obtained intervals. If λ i (π) = λ i (σ) =, or λ i (π) = λ i (σ) = and b i (π) = b i (σ) =, then for some k, a new element is added to the kth interval of both ith profiles of π and σ; this element is the largest one in the obtained intervals. If λ i (π) = λ i (σ) = and b i (π) = b i (σ), then two consecutive intervals are merged in the ith profiles of π and of σ: the kth and (k +)th ones, for some k. 8

Now we explain how the Lehmer code c c...c n is linked to the profiles of a permutation. By definition, c = and c i, i >, is the number of entries in π at the left of π i and larger than π i. If X,X,...,X p is the (i )th profile of π, it follows that c i = u j= cardx j, where u is such that u j= X j is the set of entries in π at the left of π i and larger than π i, and so c i = card u j= X j. Theorem. The mapping b: S n S n is a bijection. Proof. Let π,σ S n with b(π) = b(σ), and c c...c n and d d...d n be the Lehmer codes of π and σ. Let also i be an integer, < i n, and (I,l ),(I,l ),...,(I k,l k ) be the (i )th slice of π, and v such that π i I v (see Definition ). If X,X,...,X p is the (i )th profile of π, then if n X, it follows that c i = v j= cardx j, and if n X, it follows that c i = v j= cardx j. Since b(π) = b(σ), combining Proposition 4 and the second point of Proposition, we have that c i = d i. It follows that the Lehmer code of π and of σ are equal, and so are π and σ, and thus b is injective. And by cardinality reasons it follows that b is bijective. It is straightforward to see that the 4-tuple of statistics (Des,Lrmax,Lrmin,Rlmax) on S n has the same distribution as (Asc,Posz,Max,Rlmin) on S n. Indeed, for the Lehmer code L(π) of a permutation π we have (Des,Lrmax,Lrmin,Rlmax)π = (Asc,Posz,Max,Rlmin)L(π), see Property. But, generally, Idesπ is different from RowL(π). For example, if π = 65874, then L(π) = 464, Idesπ = {,5,7,8} and RowL(π) = {5,7,8}. Combining Theorem and Proposition it follows that b behaves not only as the Lehmer code for the above 4-tuples of statistics, but also it transforms Ides π into Row b(π). Formally, we have the next theorem, which subsequently gives Row as a set-valued partner for Asc, thereby answering to an open question stated in []. Theorem. For any π S n, (Des,Ides,Lrmax,Lrmin,Rlmax)π = (Asc,Row,Posz,Max,Rlmin)b(π), and so the multistatistic (Des,Ides,Lrmax,Lrmin,Rlmax) on S n has the same distribution as (Asc,Row,Posz,Max,Rlmin) on S n. The next corollaries are consequences of Theorem. The first of them is Foata s result in [4] saying that (asc, row) on subexcedant sequences is a double Eulerian bistatistic. Corollary. The bistatistics (asc, row) on the set of subexcedant sequences has the same distribution as (des, ides) on the set of permutations. Corollary. The bistatistics (asc, row) and (row, asc) are equidistributed on the set of subexcedant sequences. Proof. Let s S n and let define t = b(σ) where σ = π with π = b (s). It is clear that (asc,row)s = (des,ides)π = (ides,des)σ = (row,asc)t. 9

Final remarks: Recently, Lin and Kim [6] defined a bijection Ψ between two combinatorial classes counted by the large Schröder numbers, namely the set S n () of subexcedant sequences avoiding the pattern, and the set S n (4,4) of permutations avoiding the patterns 4 and 4. In [6] it is shown that Ψ proves the equidistribution of two 6-tuples of set-valued statistics, that is (Asc,Row,Posz,Max,Rlmin,Expo)s = (Des,Ides,Lrmax,Lrmin,Rlmax,Rlmin)Ψ(s), for any s S n (). Here Expo is the exposed positions statistic define in [6], and if the last statistic of both 6-tuples is dropped out, then the 5-tuples in our Theorem are obtained. Also in [6] is pointed out that the restriction of our mapping b yields a bijection between S n (4,4) and S n (), but unlike Ψ, this restriction of b does not transform Rlmin to Expo. References [] E. Aas, The double Eulerian polynomial and inversion tables, arxiv:4.565, 4. [] M. Beck, S. Robins, Computing the continuous discretely: Integer-point enumeration in polyhedra, Undergraduate Texts in Mathematics, Springer, New York, 7. [] D. Dumont, Interprétations combinatoires des nombres de Genocchi, Duke Math. J., 4() (974), 5 8. [4] D. Foata, Distributions Eulériennes et Mahoniennes sur le groupe des permutations. (French) With a comment by Richard P. Stanley. NATO Adv. Study Inst. Ser., Ser. C: Math. Phys. Sci.,, Higher combinatorics, pp. 7 49, Reidel, Dordrecht-Boston, Mass., 977. [5] D.H. Lehmer, Teaching combinatorial tricks to a computer, in Proc. Sympos. Appl. Math., (96), Amer. Math. Soc., 79 9. [6] Z. Lin, D. Kim, A sextuple equidistribution arising in pattern avoidance, arxiv:6.964v, 6. [7] R. Mantaci, F. Rakotondrajao, A permutation representation that knows what Eulerian means, Discrete Math. Theor. Comput. Sci., 4() (), 8 (electronic). [8] T.K. Petersen, Two-sided Eulerian numbers via balls in boxes, Mathematics Magazine, 86 (), 59 76. [9] C.D. Savage, M.J. Schuster, Ehrhart series of lecture hall polytopes and Eulerian polynomials for inversion sequences, J. Combin. Theory Ser. A, 9(4) () 85 87. [] M. Skandera, Dumont s statistic on words, Elec. J. of Combin., 8() (), #R. [] V. Vajnovszki, Lehmer code transforms and Mahonian statistics on permutations, Discrete Math., (), 58 589. [] M. Visontai, Some remarks on the joint distribution of descents and inverse descents, Elec. J. of Combin., () (), #P5.