Derangements with an additional restriction (and zigzags)

Transcription

1 Derangements with an additional restriction (and zigzags) István Mező Nanjing University of Information Science and Technology

2 This talk is about a class of generalized derangement numbers, and about a class of generalized zigzag numbers. The derangement part is joint work with Chen-Ying Wang (NUIST) and Piotr Miska (Jagellonian University), and based on the paper The r-derangement numbers, Discrete Mathematics, 340(7) (2017), The zigzag part is joint work with J. L. Ramírez (Universidad Nacional de Colombia).

3 The classical derangements A short introduction to the classical derangements

4 The classical derangements Let us take a permutation on n elements randomly and uniformly. What is the probability that the chosen permutation has no fixed points? (This question was raised by Pierre Raymond de Montmort (1708).) Definition: The number of fixed point free permutations on n items is denoted by D(n) and called derangement number.

5 The classical derangements The D(n) numbers satisfy a number of interesting identities. The explicit formula ) ( 1)n D(n) = n! ( ! 12! n! is valid as well as the recursion D(n) = (n 1)(D(n 1) + D(n 2)) (n 2) with the initial values D(0) = 1, D(1) = 0. Euler D(n) = nd(n 1) + ( 1) n.

6 The classical derangements By using the closed form formula, the asymptotic probability that a randomly and uniformly chosen permutation on n elements has no fixed point is D(n) lim n n! a classical and famous statement. = 1 e,

7 The classical derangements The exponential generating function of D(n) is the following: n=0 D(n) x n n! = e x 1 x.

8 The combinatorics of the r-derangement numbers The combinatorics of the r-derangement numbers

9 The combinatorics of the r-derangement numbers We define a generalization of the derangements by adding a new restriction to the definition of D n. Definition: Let 0 r n be integers. D r (n) denotes the number of derangement on n + r elements under the restriction that the first r elements are in disjoint cycles. Note: The parameter r first appeared in the theory of Stirling numbers when N. Nielsen studied the finite differences of powers at arbitrary points, and later when J. Riordan studied the connection between power moments and factorial moments. In 1984 A. Broder gave the combinatorial interpretation of Nielsen s numbers, and called these as r-stirling numbers. This is from where our motivation comes.

10 The combinatorics of the r-derangement numbers Combinatorial properties of the r-derangements

11 The combinatorics of the r-derangement numbers The permutation ( ) is a derangement, but not a 2-derangement: (1 3 2), and (4 5 6) are the two cycles of the permutation; 1 and 2 belong to the same cycle.

12 The combinatorics of the r-derangement numbers The permutation ( ) is already a permitted 2-derangement. It has two cycles: (1 3 4)(2 5 6), and the first two elements are in distinct cycles.

13 The combinatorics of the r-derangement numbers Recursion for D r (n) For all n > 2 and r > 0 we have that D r (n) = rd r 1 (n 1) + (n 1)D r (n 2) + (n + r 1)D r (n 1). The initial conditions are the followings: D 1 (n) = D(n + 1), D r (r) = r! (r 1), D r (r + 1) = r(r + 1)! (r 2). Note that needs values from lower r parameter.

14 The combinatorics of the r-derangement numbers Another recursion for D r (n) For all n > 0 and r 0 D r+1 (n) = n r r + 1 D r (n) + n r + 1 D r (n 1). This is a straight generalization of the de Montmort recurrence D(n + 1) = n(d(n) + nd(n 1)). (However, our proof is algebraic.)

15 The combinatorics of the r-derangement numbers The first members of the sequence D 2 (n) (n 2) are 2, 12, 84, 640, 5 430, , , , , ,... while the first members of D 3 (n) starting from n = 3 are 6, 72, 780, 8 520, , , , , ,...

16 The combinatorics of the r-derangement numbers The exponential generating function D r (n) D r (n) x n n! = x r e x (1 x) r+1. n=0 Note that r = 0 gives back the D(n) case.

17 The combinatorics of the r-derangement numbers There are many formulas for the r-derangement numbers. (One more parameter gives one more degree of freedom to play with!) Identities for D r (n) part I Let r 1 and s {1,..., r}. Then for each n s we have D r (n) = n j=s ( ) j 1 n! s 1 (n j)! D r s(n j). In particular, D r (n) = n j=r ( ) j 1 n! r 1 D(n j), n r. (n j)!

18 The combinatorics of the r-derangement numbers Identities for D r (n) part II Additionally, we have a closed formula for the r-derangement numbers: n ( ) j ( 1) n j D r (n) = n! r (n j)!, n r. j=r

19 The combinatorics of the r-derangement numbers A connection with the Lah numbers can also be given. The n k number counts the partitions of n elements into k blocks such that the order of the elements in the individual blocks count, but the order of the blocks is not taken into account. Such partitions are often called ordered lists. The following identity is valid: n (r + 1)! r + 1 = n k=1 ( ) n k D r (n k). k

20 Asymptotics of the r-derangements Asymptotics of the r-derangements

21 Asymptotics of the r-derangements Asymptotics of the r-derangements (Note that r=0 1 1 r! lim n e = 1!) D r (n) (n + r)! = 1 1 r! e.

22 Asymptotics of the r-derangements Actually, much better asymptotic approximations can be given by the saddle point method. Let A(r, k) = r i=0 Then we have the following statement. ( 1) i ( ) r. (k i)! i Asymptotics of the r-derangements refined version D r (n) n! for any ε > 0. = ( 1)n e r ( ) k 1 r A(r, k) + O(ε n ) n k=0

23 Asymptotics of the r-derangements In particular, for r = 2: and for r = 3: D 2 (n) (n + 2)! = 1 2e D 3 (n) (n + 3)! = 1 6e n 2 + n 1 (n + 1)(n + 2) + O(εn ) 1 1 2! e, n 3 4n + 1 (n + 1)(n + 2)(n + 3) + O(εn ) 1 1 3! e.

24 Asymptotics of the r-derangements We note that these approximations are rather close even for small values of n. For example, D 4 (8) (8 + 4)! = , while from the above approximation we get the estimation D 4 (8) (8 + 4)! Nine digits already agree for as small n as n = 8! (Here! is not the factorial but the exclamation mark. :))

25 Some number theoretical connections Some number theoretical connections

26 Some number theoretical connections There are some nice number theoretical facts about the r-derangements. For example, take a look at the last digits of D 2 (n): 0, 0, 2, 2, 4, 0, 0, 8, 8, 6, 0, 0, 2, 2, 4, 0, 0, 8, 8, 6,... They seem to be periodic of length 10!

27 Some number theoretical connections What about D 3 (n)? 0, 0, 0, 6, 2, 0, 0, 0, 4, 8, 0, 0, 0, 6, 2, 0, 0, 0, 4, 8,... Again, seems to be periodic with the same length. We may have the following conjecture: D r (n + 10) D r (n) (mod 10).

28 Some number theoretical connections We managed to prove a much more general statement. Periodicity property Let r, d 1 arbitrary. If d is even, then D r (n + d) D r (n) (mod d), and if d is odd, then D r (n + 2d) D r (n) (mod d). If, in particular, d = 10 then D r (n + 10) D r (n) (mod 10) indeed holds as we had conjectured.

29 Some number theoretical connections Some more number theoretical properties of the r-derangements can be proven. First we make an observation: n! (n r)! D r (n), n! thus it is worth to factor out (n r)!, and define C r (n) = (n r)! D r (n). n!

30 Some number theoretical connections It is worth to ask, how many prime divisors have the sequence (C r (n)) n=0 if r is fixed. The answer is in the following statement. Theorem There are infinitely many prime divisors of the set for any fixed r > 0. Conjecture {C r (n) n 0}, Let B r be the (infinite) set of prime divisors of the above set, and let A r = P \ B r. Then A r is also infinite.

31 Some number theoretical connections A stronger conjecture is actually made by P. Miska: Conjecture Let A r as before. Then A r {1,..., n} lim n P {1,..., n} = 1 e. Some heuristic arguments seem to support this conjecture.

32 Some number theoretical connections It was proven by P. Miska, that the diophantine equation D(n) = q m! has only finitely many solutions (q fixed rational number, n, m are positive integer variables). In the case when r > 1, the situation is more complicated, we need knowledge on the set A r. What we have is the following: Assume that r 2, and A r. Then for any q Q the diophantine equation D r (n) = q m! has only finitely many solutions, and the bound can be explicitly determined for m.

33 Some number theoretical connections Pure prime power r-derangements The only solution to the diophantine equation D r (n) = p k is (n, r, p, k) = (2, 2, 2, 1).

34 The classical zigzag numbers Zigzag permutations and a generalization

35 The classical zigzag numbers Definition A permutation π is called zigzag or down-up alternating if π 1 > π 2 < π 3 > π 4. On n letters there are E n zigzags, these numbers are the Euler numbers (A000111). For example, 4132 is a zigzag permutation.

36 The classical zigzag numbers A classical result is the following André (1881) The exponential generating function of the Euler zigzag numbers is n=0 E n x n n! = tan x + sec x.

37 The r-zigzag numbers The r-zigzag permutations

38 The r-zigzag numbers Our goal is to introduce the parameter r which, as we have seen, restricts the first r elements from sharing cycles. But how to do this in zigzag permutations? Zigzagness and cycle structures are not related. We have a way out: we recall the standard form of a cycle decomposition of a permutation: 1) by cyclic shifts put the minimal elements into the first position in each cycle (left-to-right minima), 2) order the cycles in decreasing order with respect to their first (and so minimal) elements.

39 The r-zigzag numbers We can now easily see that when the first r elements are in different cycles the standard from results in a permutation where 1, 2,..., r are left-to-right minima (and vice versa). Thus, requiring the first r elements to be in different cycles in a zigzag is the same as requiring the following. Definition An r-zigzag (or r-down-up-alternating) permutation is a permutation on n + r elements when it is zigzag, and the first r elements are left-to-right minima. The number of r-zigzag permutations is denoted by E n,r. We obviously have that E n,0 = E n, and E n,1 = E n+1.

40 The r-zigzag numbers A 2-zigzag on 6 elements is (It is also a 3-zigzag, but not a 4-zigzag.)

41 The r-zigzag numbers André s results generalizes to the r-zigzags in the following way: Ramírez-M. The exponential generating function of the r-zigzag numbers, when r 2, is n=0 E n,r x n n! = (tanr (x) + tan r 2 x)(tan x + sec x).

42 The r-zigzag numbers The bi-variate generating function is also determined (easily from the former result): Ramírez-M. r=2 n=0 x n E n,r n! y r = y 2 (tan x + sec x) sec 2 x. 1 y tan x

43 The r-zigzag numbers The asymptotic behavior of E n,r as n and r 2 fixed is given: If r 2, then E n,r n! 2 ( 2 π as n tends to infinity. ) n+r+1 ( ) ( n + r 11 + r 6 2r 3 ) ( 2 π ) n+r 1 ( ) n + r 2 r 2

44 The r-zigzag numbers The E n,2 sequence appears in OEIS under the ID number A There it is written that A is the number of zigzag permutations when the number n appears on the left of 1. These permutations were studied by Callan and Heneghan-Petersen, and they call them max-min. Ramírez-M. There exists a bijection between the 2-zigzag permutations on n elements and the max-min zigzag permutations on n + 2 elements.

45 An application of the r-zigzag numbers An application of the r-zigzag numbers

46 An application of the r-zigzag numbers The tangent and secant numbers are defined by their Taylor series: x n T n = tan x, n! n=0 x n S n = sec x. n! n=0 The higher order tangent and secant numbers T n (k) defined by the Taylor series n=0 n=0 T n (k) x n n! = tank x, S n (k) x n n! = sec x tank x. and S (k) n These appeared first in an 1972 paper of Carlitz and Scoville. are

47 An application of the r-zigzag numbers These numbers count up-down zigzags such that we adjoin k infinite elements which are bigger than any of the n elements. A typical permutation of this kind is

48 An application of the r-zigzag numbers The higher order tangent and secant numbers and the r-zigzag numbers are closely related. For all n 0 and r 2 E n,r = n k=0 ( ) n E k+1 S (r 2) k n k. For all n 0 E n,r = { S n (r) T n (r+1) + S n (r 2), if n r (mod 2); + T n (r 1), if n r (mod 2).

49 An application of the r-zigzag numbers Thank you for your attention!