INVOLUTIONS AND THEIR PROGENIES TEWODROS AMDEBERHAN AND VICTOR H. MOLL Abstract. Any permutation has a disjoint cycle decomposition and concept generates an equivalence class on the symmetry group called the cycle-type. The main focus of this work is on permutations of restricted cycle-types, with particular emphasis on the special class of involutions and their partial sums. The paper provides generating functions, determinantal expressions, asymp- totic estimates as well as arithmetic and combinatorial properties. 1. Introduction For n N, the group of permutations in n symbols a , a , ,a is called the ∈ { 1 2 ··· n} symmetric group, denoted by Sn. A cycle ρ Sn is a permutation of the form, in a one-line notation, ρ = (a a a ). The∈ notation indicates that all the entries i1 i2 ··· ir of the cycle are distinct and ρ(aij )= aij+1 for 1 j r 1 and ρ(air )= ai1 . The cycle ρ is said to have length r, written as r = ≤L(ρ).≤ Every− permutation π S ∈ n can be written as a product of cycles π = ρ1ρ2 ρm. This decomposition is not unique, but if the cycles are assumed to be disjoint··· and the lengths are taken in weakly decreasing order, then L(ρ1),L(ρ2), ,L(ρm) is uniquely determined by π, called the cycle type of π. { ··· } The following notation is used: for 1 ℓ, t n, ≤ ≤ (1.1) C = π S with every cycle in π of length at most ℓ , n,ℓ { ∈ n } (1.2) α (π) = number of cycles in π S of length t. t ∈ n and the cardinality of Cn,ℓ is denoted by dn,ℓ = #Cn,ℓ. 2 Definition 1.1. A permutation π in Sn is called an involution if π (j) = j, for 1 j n. The set of involutions in S is denoted by Inv(n). The cardinality of ≤ ≤ n arXiv:1406.2356v1 [math.CO] 9 Jun 2014 this set, denoted by I1(n), is called the involution number. The factorization of π as a product of disjoint cycles shows that any cycle in the factorization of an involution has length 1 or 2. This implies Inv(n)= Cn,2 and thus I (n)= d . It follows that if π Inv(n) is an involution, then α (π)+2α (π)= n. 1 n,2 ∈ 1 2 Example 1.2. Every permutation of 2 symbols (a transposition) is an involution and for n = 3 there are 4 involutions (1.3) π1 = (1)(2)(3), π2 = (12), π3 = (13), π4 = (23). The cycles (123) and (132) are the only elements of S3 that are not involutions. Therefore I1(2) = 2 and I1(3) = 4. Date: October 16, 2018. 1991 Mathematics Subject Classification. Primary 05A15, 11B75. Key words and phrases. involutions, valuations, asymptotics. 1 2 TEWODROSAMDEBERHANANDVICTORH.MOLL Elementary properties of the numbers I1(n) are described in Section 2. These include a second order recurrence, an exponential generating function as well as an explicit finite sum. These are generalized to the involution polynomials I1(n; t) in Section 3 which are intimately linked to the (probabilistic) Hermite polynomials defined by ⌊n/2⌋ ( 1)j tn−2j (1.4) H (t)= n! − n j!(n 2j)! 2j j=0 X − with generating function ∞ xn (1.5) H (t) = exp xt 1 x2 . n n! − 2 n=0 X The involution polynomials have a combinatorial interpretation as the generating function for fixed points of permutation in Sn. Arithmetic properties of I1(n) are presented in Section 4. Particular emphasis is given to the 2-adic valuation of I1(n). Recall that, for x N and p prime, the p-adic valuation of x, denoted by νp(x), is the highest power∈ of p that divides x. An odd prime p is called efficient if p does not divide I (j) for 0 j p 1. Otherwise it is called inefficient. The 1 ≤ ≤ − prime p = 3 is efficient and p = 5 is inefficient since I1(4) = 10. A periodicity argument is used to show that νp(I1(n) = 0; i.e., p is efficient. Morever, for a prime p, it is shown that either p divides I1(n) infinitely often or never. In the case of an inefficient prime, it is conjecture that the p-adic valuation of the sequence I1(n) can be given in terms of a tree Tp. This phenomena is illustrated for the prime p = 5. It is an open question to characterize efficient (or inefficient) primes. The partial sums of I1(n), denoted by an, are discussed in Section 5. Their arithmetic properties are presented in Section 6. For instance, an explicit expression for their 2-adic valuation is given there. The valuations for odd primes are also conjectured to have a tree structure. This is illustrated in the case p = 5. Section 7 considers the statistics of the sequence dn,ℓ in (1.1). This is a generalization of I1(n)= Cn,2. Finally, the asymptotic behavior of d = C is given in Section 8. n,ℓ | n,ℓ| 2. Basic properties of the involution numbers This section discusses fundamental properties of I1(n). Some of them are well- known but proofs are included here for the convenience of the reader. Theorem 2.1. The sequence I1(n) satisfies the recurrence (2.1) I (n)= I (n 1)+(n 1)I (n 2), for n 2, 1 1 − − 1 − ≥ with initial conditions I1(0) = I1(1) = 1. Proof. There are I1(n 1) involutions that fix n. The number of involutions that contain a cycle (jn), with− 1 j n 1 is n 1 times the number of involutions containing the cycle (n 1, n≤). This≤ is− (n 1)−I (n 2). − − 1 − The recurrence above generates the values n 0 1 2 3 4 5 6 7 8 9 10 I1(n) 1 1 2 4 10 26 76 232 764 2620 9496 This is sequence A000085 in OEIS. The recurrence (2.1) now enables to write a generating function for I (n) . { 1 } INVOLUTIONSANDTHEIRPROGENIES 3 Theorem 2.2. The exponential generating function for I1(n) is ∞ I (n) (2.2) 1 xn = exp(x + 1 x2). n! 2 n=0 X Proof. On the basis of (2.1) verify that both sides of (2.2) satisfy f ′(x)=(1+ x)f(x) and the value f(0) = 1. x x2/2 Cauchy’s product formula on e and e allows to express I1(n) as a finite sum. Corollary 2.3. The involution numbers I1(n) are given by ⌊n/2⌋ n 2j j! (2.3) I (n)= . 1 2j j 2j j=0 X 2j j! The numbers appearing in (2.3) are now shown to be of the same parity. j 2j Corollary 2.4. For j N, the numbers (2j)!/(j!2j) are odd integers. ∈ Proof. The identity (2j)! (2j)(2j 1) (j + 1) (2.4) = − ··· j!2j 2j shows that the denominator is a power of 2. To compute this power, use Legendre’s formula (2.5) ν (n!) = n s (n), 2 − 2 where s2(n) is the sum of the digits of n in its binary expansion. Therefore, (2j)! (2.6) ν = (2j s (2j)) (j s (j)) j =0, 2 j!2j − 2 − − 2 − in view of s2(2j)= s2(j). A second recurrence for the involution numbers is presented next. Theorem 2.5. For n,m N, the involution numbers satisfy ∈ n m (2.7) I (n + m)= k! I (n k)I (m k). 1 k k 1 − 1 − kX≥0 Proof. Split up the set [n+m] into two disjoint subsets A and B, of n and m letters, respectively. Count the involutions in Inv(n + m) acoording to the number k of cross-permutations that make up a cycle (ab), with a A and b B. The letters n m ∈ ∈ a and b can be chosen in k k ways and k! ways to place the k cycles (ab). The remaining elements in A (respectively B) allow I1(n k) (respectively I1(m k)) n− m − involutions. To complete the argument, summing k! k k I1(n k)I1(m k) over k. − − As a direct consequence of (in fact, equivalent to) Theorem 2.5 the following analytic statement is recorded. This result bypasses the need for an otherwise messy chain rule for derivatives. 4 TEWODROSAMDEBERHANANDVICTORH.MOLL Corollary 2.6. Higher order derivatives of the function f(x) = exp x + x2/2 are computed by the umbral dm m m (2.8) f(x)= f(x) I (m k)xk := f(x)(x + I )m. dxm k 1 − 1 Xk=0 dm xn Proof. From Theorem 2.2, f(x) = I (n + m) . The right-hand side of dxm 1 n! n Theorem 2.5 implies X n m xn k! I (n k)I (m k) = k k 1 − 1 − n! n X Xk m xn−k I (m k)xk I (n k) k 1 − 1 − (n k)! n Xk X − m = f(x) I (m k)xk. k 1 − Xk The claim follows. r The recurrence (2.7) is now used to prove periodicity of I1(n) mod p . r Theorem 2.7. Let p be a prime and r N. Then I1 mod p is a periodic sequence of period pr. ∈ Proof. Write n = cpr + t with 0 t<pr. Theorem 2.5 gives ≤ t cpr t (2.9) I (cpr + t)= k! I (cpr k)I (t k). 1 k k 1 − 1 − Xk=0 r For k> 0, cp k! = (cpr)(cpr 1) (cpr k + 1) 0 mod pr yields k − ··· − ≡ r r r (2.10) I1(cp + t) I1(cp )I1(t) mod p .