c Birkh¨auser Verlag, Basel, 2001 Annals of Combinatorics 5 (2001) 479-491 0218-0006/01/040479-13$1.50+0.20/0 Annals of Combinatorics Generalized Riffle Shuffles and Quasisymmetric Functions Richard P. Stanley∗ Department of Mathematics 2-375, Massachusetts Institute of Technology, Cambridge, MA 02139, USA [email protected] Received March 28, 2001 AMS Subject Classification: 05B05, 05A05, 60C05 In memory of Gian-Carlo Rota: adviser, mentor, colleague and friend Abstract. Given a probability distribution on a totally ordered set, we define for each n 1 a ≥ related distribution on the symmetric group Sn, called the QS-distribution. It is a generalization of the q-shuffle distribution considered by Bayer, Diaconis, and Fulman. The QS-distribution is closely related to the theory of quasisymmetric functions and symmetric functions. We obtain explicit formulas in terms of quasisymmetric and symmetric functions for the probability that a random permutation from the QS-distribution satisfies various properties, such as having a given descent set, cycle structure, or shape under the Robinson-Schensted-Knuth algorithm. Keywords: QS-distribution, q-shuffle, quasisymmetric function, major index, RSK algorithm, increasing subsequence 1. Introduction Let xi be a probability distribution on a totally ordered set I, i.e., the probability of i I is xi. (Hence xi 0 and ∑xi = 1.) Fix n P = 1; 2;::: , and define a random 2 ≥ 2 f g permutation w Sn as follows. For each 1 j n, choose independently an integer 2 ≤ ≤ i j (from the distribution xi). Then standardize the sequence ii = i1 in in the sense ··· of [34, p. 322], i.e., let α1 < < αk be the elements of I actually appearing in ii, and ··· let ai be the number of αi’s in i. Replace the α1’s in i by 1; 2;:::;a1 from left-to-right, then the α2’s in i by a1 + 1; a1 + 2;:::;a1 + a2 from left-to-right, etc. For instance, if I = P and i = 311431, then w = 412653. This defines a probability distribution on the symmetric group Sn, which we call the QS-distribution (because of the close connection with quasisymmetric functions explained below). If we need to be explicit about the parameters x = (xi)i I , then we will refer to the QS(x)-distribution. 2 ∗ Partially supported by NSF grant DMS-9988459. 479 480 R.P. Stanley The QS-distribution is not new, at least when I is finite. It appears for instance in [10, pp. 153–154] [14,20,24]. Although these authors anticipate some of our results, they don’t make systematic use of quasisymmetric and symmetric functions. Some additional work using the viewpoint of our paper is due to Fulman [15]. Example 1.1. As an example of a general result to be proved later (Theorem 2.1), let us compute Prob(213), the probability that a random permutation w S3 (chosen from 2 the QS-distribution) is equal to the permutation 213. A sequence ii = i1i2i3 will have the standardization 213 if and only if i2 < i1 i3. Hence ≤ Prob(213) = ∑ xaxbxc: a<b c ≤ There is an alternative way to describe the QS-distribution. Suppose we have a deck of n cards. Cut the deck into random packets of respective sizes ai > 0; i I, such that 2 ∑ai = n and the probability of (ai)i I is 2 ai xi Prob((ai)i I ) = n!∏ : 2 i I ai! 2 Then riffle shuffle these packets Pi together into a pile P in the manner described by Bayer and Diaconis [2] (see also [11, 14]); namely, after placing k cards on P, the probability that the next card comes from the top of the current packet Pj is proportional to the current number of cards in Pj. This card is then placed at the bottom of P. The ordinary dovetail or riffle shuffle [2, p. 294] corresponds to the case I = 1; 2 and x1 = f g x2 = 1=2. In this case the original deck is cut according to the binomial distribution. More generally, if for fixed q P we cut the deck into q packets (some possibly empty) according to the q-multinomial2 distribution, then we obtain the q-shuffles of Bayer and Diaconis [2, p. 299] (where they use a for our q). The q-shuffle is identical to the QS-distribution for I = 1; 2:::;q and x1 = x2 = = xq = 1=q. We will denote this f g ··· distribution by Uq. If we q-shuffle a deck and then r-shuffle it, the distribution is the same as a single qr-shuffle [2, Lemma 1]. In other words, if denotes convolution of ∗ probability distributions then Uq Ur = Uqr. We extend this result to the QS-distribution in Theorem 2.4. ∗ In the next section we will establish the connection between the QS-distribution and the theory of quasisymmetric functions. In Section 3 we use known results from the theory of quasisymmetric and symmetric functions to obtain results about the QS- distribution, many of them direct generalizations of the work of Bayer, Diaconis, and Fulman mentioned above. For instance, we show that the probability that a random permutation w Sn (always assumed chosen from the QS-distribution) has k inversions is equal to the probability2 that w has major index k. This is an analogue of MacMahon’s corresponding result for the uniform distribution on Sn (see e.g., [33, p. 23]). A further result is that if T 0 is a standard Young tableau of shape λ, then the probability that T 0 is the recording tableau of w under the Robinson-Schensted-Knuth algorithm is sλ(x), where x = (xi)i I and sλ denotes a Schur function. 2 Generalized Riffle Shuffles 481 2. The QS-Distribution and Quasisymmetric Functions In this section we show the connection between the QS-distribution and quasisymmet- ric functions. In general our reference for quasisymmetric functions and symmetric functions will be [34, Ch. 7]. Quasisymmetric functions may be regarded as bearing the same relation to compositions (ordered partitions of a nonnegative integer) as sym- metric functions bear to partitions. Namely, the homogeneous symmetric functions of degree n form a vector space of dimension p(n) (the number of partitions of n) and have many natural bases indexed by the partitions of n. Similarly, the homogeneous n 1 quasisymmetric functions of degree n form a vector space of dimension 2 − (the num- ber of compositions of n) and have many natural bases indexed by the compositions of n. A quasisymmetric function F(z) is a power series of bounded degree, say with ra- tional coefficients, in the variables z = (zi)i I, with the following property: let i1 < i2 < 2 < in and j1 < j2 < < jn, where ik; jk I. If a1;:::;ak P, then the coefficient of z··a1· zan is equal to the···coefficient of za1 2zan in F(z). The2set of all quasisymmetric i1 in j1 jn functions··· forms a graded Q-algebra denoted··· Q . The quasisymmetric functions that are homogeneous of degree n form a Q-vector space, denoted Qn. If I n (including n 1 j j ≥ I = ∞), then dimQn = 2 . j j − Let α = (α1;:::;αk) be a composition of n, i.e., αi P and ∑αi = n. Let Comp(n) 2 n 1 denote the set of all compositions of n, so #Comp(n) = 2 − [33, pp. 14–15]. Define Sα = α1; α1 + α2;:::;α1 + + αk 1 1;:::;n 1 : f ··· − g ⊆ f − g The fundamental quasisymmetric function Lα = Lα(z) is defined by Lα(z) = ∑ zi1 zin : i in ··· 1≤···≤ i <i if j Sα j j+1 2 It is not hard to show that when I n, the set Lα : α Comp(n) is a Q-basis for Qn j j ≥ f 2 g [34, Prop. 7.19.1]. More generally, if I = k then the set Lα : α Comp(n); length(α) j j f 2 k is a Q-basis for Qn, and Lα = 0 if length(α) > k. ≤ g If w = w1w2 wn Sn then let D(w) denote the descent set of w, i.e., ··· 2 D(w) = i : wi > wi+1 : f g Write co(w) for the unique composition of n satisfying Sco(w) = D(w). In other words, if co(w) = (α1;:::;αk), then w has descents exactly at α1, α1 +α2;:::, α1 + +αk 1. ··· − For any composition α of n it is easy to see that the (possibly infinite) series Lα(x) (where as always xi 0 and ∑xi = 1) is absolutely convergent and therefore represents a nonnegative real number≥ . The main result on which all our other results depend is the following. Theorem 2.1. Let w Sn. The probability Prob(w) that a permutation in Sn chosen from the QS-distribution2 is equal to w is given by Prob(w) = L 1 (x): co(w− ) 482 R.P. Stanley Proof. This result is equivalent to [18, Lemma 3.2] (see also [30, Lemma 9.39]). Be- cause of its importance here we present the (easy) proof. The integers i for which i + 1 1 appears somewhere to the left of i in w = w1w2 wn are just the elements of D(w ). ··· − Let i1;:::;in be chosen independently from the distribution xi. Let a j = i 1 , i.e., if w− ( j) we write down i1 in underneath w1 wn, then a j appears below j. In order for the ··· ··· sequence i1 in to standardize to w it is necessary and sufficient that a1 a2 an ··· 1 ≤ ≤ ··· ≤ and that a j < a j+1 whenever j D(w ).