Some Enumerations for Parking Functions

In this paper, let Pn,n+k;≤n+k (resp. Pn;≤s) denote the set of parking functions α = (a1, · · · , an) of length n with n+k (respe. n)parking spaces satisfying 1 ≤ ai ≤ n+k (resp. 1 ≤ ai ≤ s) for all i. Let pn,n+k;≤n+k = |Pn,n+k;≤n+k| and pn;≤s = |Pn;≤s|. Let P l n;≤s denote the set of parking functions α = (a1, · · · , an) ∈ Pn;≤s such that a1 = l and pn;≤s = |P l n;≤s|. We derive some formulas and recurrence relations for the sequences pn,n+k;≤n+k, pn;≤s and p l n;≤s and give the generating functions for these sequences. We also study the asymptotic behavior for these sequences. Keyword: parking function; leading term; asymptotic behavior Partially supported by NSC 96-2115-M-006-012 Email address of the corresponding author: [email protected] [email protected] 1 Introductioin Throughout the paper, we let [n] := {1, 2, · · · , n} and [m,n] := {m, · · · , n}. Suppose that n cars have to be parked in m parking spaces which are arranged in a line and numbered 1 to m from left to right. Each car has initial parking preference ai; if space ai is occupied, the car moves to the first unoccupied space to the right. We call (a1, · · · , an) preference set. Clearly, the number of preference sets is m. If a preference set (a1, · · · , an) satisfies ai ≤ ai+1 for 1 ≤ i ≤ n−1, then we say that this preference set is ordered. If all the cars can find a parking space, then we say the preference set is a parking function. If there are exactly k cars which can’t be parked, then the preference set is called a k-flaw preference set. Let n, m, s, and k be four nonnegative integers with 1 ≤ s ≤ m and k ≤ n − 1. Suppose there are m parking spaces. We use Pn,m;≤s;k to denote a set of k-flaw preference sets (a1, · · · , an) of length n satisfying 1 ≤ ai ≤ s for all i. For 1 ≤ l ≤ s, we use P l n,m;≤s;k to denote a set of preference sets (a1, · · · , an) ∈ Pn,m;≤s;k such that a1 = l. Let Pn,m;=s;k (resp. P l n,m;=s;k) be a set of preference sets (a1, · · · , an) ∈ Pn,m;≤s;k(resp. ∈ P l n,m;≤s;k) such that aj = s for some j. Let pn,m;≤s;k = |Pn,m;≤s;k|, p l n,m;≤s;k = |P l n,m;≤s;k|, pn,m;=s;k = |Pn,m;=s;k| and pln,m;=s;k = |P l n,m;=s;k|. For any of the above cases, if the parameter k ( resp. m ) doesn’t appear, we understand k = 0 ( resp. m = n ); if the parameter m and s are both erased, we understand s = m = n. There are some results about parking functions with s = m = n. Riordan introduced parking functions in [11]. He derived that the number of parking functions of length n is (n+ 1), which coincides with the number of labeled trees on n+1 vertices by Cayley’s formula. Several bijections between the two sets are known (e.g., see [5, 11, 12]). Furthermore, define a generating function P (x) = ∑ n≥0 (n+1) n! x. It is well known that xP (x) is the compositional inverse of the function ψ(x) = xe, i.e., ψ(xP (x)) = x. Riordan concluded that the number of ordered parking functions is 1 n+1 ( 2n n ) , which is also equals the number of Dyck path of semilength n. Parking functions have been found in connection to many other combinatorial structures such as acyclic mappings, polytopes, non-crossing partitions, non-nesting partitions, hyperplane arrangements,etc. Refer to [4, 5, 6, 10, 13, 14] for more information. 2 Any parking function (a1, · · · , an) can be redefined that its increasing rearrangement (b1, · · · , bn) satisfies bi ≤ i. Pitman and Stanley generalized the notion of parking functions in [10]. Let x = (x1, · · · , xn) be a sequence of positive integers. The sequence α = (a1, · · · , an) is called an x-parking function if the non-decreasing rearrangement (b1, · · · , bn) of α satisfies bi ≤ x1 + · · · + xi for any 1 ≤ i ≤ n. Thus, the ordinary parking function is the case x = (1, · · · , 1). By the determinant formula of Gončarove polynomials, Kung and Yan [9] obtained the number of x-parking functions for an arbitrary x. See also [15, 16, 17] for the explicit formulas and properties for some specified cases of x. An x-parking function (a1, · · · , an) is said to be k-leading if a1 = k. Let qn,k denote the number of k-leading ordinary parking functions of length n. Foata and Riordan [5] derived a generating function for qn,k algebraically. Recently, Sen-peng Eu, Tung-shan Fu and Chun-Ju Lai [1] gave a combinatorial approach to the enumeration of (a, b, · · · , b)-parking functions by their leading terms. Riordan [11] told us the relations between ordered parking functions and Dyck paths. Senpeng Eu et al. [2, 3] considered the problem of the enumerations of lattice paths with flaws. It is natural to consider the problem of the enumerations of preference sets with flaws. Ordered k-flaw preference sets were studied in [7]. Building on work in this paper, we give enumerations of k-flaw preference sets in [8]. In this paper, we first consider enumerations of parking functions in Pn,m;≤m. When m ≥ n, Riordan [11] gave a explicit formula pn,m;≤m = (m − n + 1)(m + 1) . We obtain another formula pn,n+k;≤n+k = ∑ r0+···+rk=n ( n r0,··· ,rk ) k ∏ i=0 (ri + 1) ri−1 for any n ≥ 0 and k ≥ 0 and find that the sequence pn,n+k;≤n+k satisfies the recurrence relation pn,n+k;≤n+k = k ∑ i=0 ( n i ) pipn−i,n−i+k−1;≤n−i+k−1. When m < n, at least n−m cars can’t find parking spaces. We conclude that pn,m;≤m;n−m = m n − m−2 ∑ i=0 ( n i ) (i+ 1)(m− i− 1) for any 0 ≤ m ≤ n. Then, we focus on the problem of enumerations of parking functions in Pn;≤s. We prove that pn;≤s = pn,s;≤s;n−s by a bijection from the sets Pn;≤s to Pn,s;≤s;n−s for any 1 ≤ s ≤ n. Also we obtain that pn;≤n−k = k+1 ∑ i=0 (−1) ( n i ) (n−i+1)(k+1−i) for any 0 ≤ k ≤ n−1. Furthermore, for any n ≥ k+1, we derive two recurrence relations pn;≤n−k = pn;≤n−k+1− k ∑ i=1 ( n i ) pn−i;≤n−k and 3 pn;≤n−k = (n+1) − k ∑ i=1 ( n i ) (k−i+1)(k+1)pn−i;≤n−k. Since pn;=n−k+1 = pn;≤n−k+1−pn;≤n−k, we have pn;=n−k = pn;=n−k+1+ ( n k+1 ) (n−k)− k ∑ i=1 ( n i ) pn−i;=n−k for any k ≥ 1 and n ≥ k+1, with pn;=n = n . Motivated by the work of Foata and Riordan in [5] as well as Sen-Peng Eu et al. in [1], we investigate the problem of the enumerations of some parking functions with leading term l. We derive the formula pn;≤s = s − l−2 ∑ i=0 ( n−1 i ) (s−i−1)pi− s−2 ∑ i=l ( n−1 i−1 ) (s−i−1)pi for any 1 ≤ l ≤ s ≤ n. We prove that pn;≤s = pn−1;≤s by a bijection from the sets P s n;≤s to Pn−1;≤s for any 1 ≤ s ≤ n− 1. Furthermore, for any n ≥ k+1 and l ≤ n− k, we conclude that pln;≤n−k = pln;≤n−k+1− k ∑ i=1 ( n−1 i ) pln−i;≤n−k and p l n;≤n−k = p l n− k ∑ i=1 ( n−1 i ) (k− i+1)(k+1)pn−i;≤n−k. Noting that pln;=n−k+1 = p l n;≤n−k+1 − p l n;≤n−k, we obtain p n n;=n = pn−1 and p n−k n;=n−k = pn−1;≤n−k for any k ≥ 1. Let k ≥ 1, then pln;=n−k = p l n;=n−k+1+ ( n−1 k+1 ) pln−k−1− k ∑ i=1 ( n−1 i ) pln−i;=n−k for any n ≥ k+2 and l ≤ n− k − 1. Also we give the generating functions of some sequences. For a fixed k ≥ 0, we define a generating function Qk(x) = ∑ n≥0 pn,n+k;≤n+k n! x, then Q0(x) = P (x), Qk(x) = Qk−1(x)P (x) for any k ≥ 1. Let Q(x, y) = ∑ k≥0 Qk(x)y , then Q(x, y) = P (x) 1−yP (x) . Let Rk(x) = ∑ n≥k pn;≤n−k n! x for any k ≥ 0, then R0(x) = P (x) and Rk+1(x) = Rk(x) − k+1 ∑ i=1 x i! Rk+1−i(x) for any k ≥ 1, with initial condition R1(x) = (1− x)P (x)− 1. Using this recurrence relation, by induction, we prove that Rk(x) = P (x) k ∑ i=0 (−1)(k+1−i) i! x− k−1 ∑ i=0 (−1)(k−i) i! x for any k ≥ 0. Let R(x, y) = ∑ k≥0 Rk(x)y , then R(x, y) = P (x)−y exy−y . Let Hk(x) = ∑ n≥k pn;=n−k n! x, then Hk(x) = Hk−1(x) + x (k+1)! P (x) − k ∑ i=1 x i! Hk−i(x) for any k ≥ 2, with initial conditions H0(x) = xP (x) + 1 and H1(x) = P (x)(x− 1 2 x)− x. In fact, we may prove that for any k ≥ 0, Hk(x) = P (x) [ k ∑ i=0 (−1)(k + 1− i) i! x − k+1 ∑ i=0 (−1)(k + 2− i) i! x ]