A recovery of two determinantal representations for derangement numbers

In the paper, the authors recover, correct, and extend two representations for derangement numbers in terms of a tridiagonal determinant. Subjects: Advanced Mathematics; Analysis Mathematics; Calculus; Combinatorics; DiscreteMathematics; Mathematics & Statistics; Number Theory; Real Functions; Science; Special Functions


Introduction
In combinatorics, a derangement is a permutation of the elements of a set, such that no element appears in its original position. The number of derangements of a set of size n is called the derangement number and sometimes denoted by !n. The problem of counting derangements was first considered in 1708 and solved in 1713 by Pierre Raymond de Montmort, as did Nicholas Bernoulli at about the same time. Derangement numbers !n arise naturally in many different contexts. More generally, the number of derangements in various families of transitive permutation groups has been studied extensively in recent years. For more information on !n, please refer to Aigner (2007), Andreescu and Feng (2004), Wilf (1994Wilf ( , 2006

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A derangement is a permutation of elements of a set, such that no element appears in its original position. The number of derangements of a set is called the derangement number. The problem of counting derangements was first considered in 1708 and solved in 1713. Derangement numbers arise naturally in many different contexts. The number of derangements in various families of transitive permutation groups has been studied extensively in recent years.
In the paper, by virtue of an old formula for computing derivatives of a ratio between two differentiable functions in terms of the Hessenberg determinants, the authors recover, correct, and extend two representations for derangement numbers in terms of a tridiagonal determinant.

Proof of Theorem 1
Now we are in a position to prove Theorem 1.
Applying u(x) = e x and v(x) = 1 + x in Lemma 1 gives as x → 0 for 1 ≤ k ≤ n + 1 and as x → 0 for 1 ≤ i ≤ n + 1 and 1 ≤ j ≤ n. Consequently, employing (6) reveals (4), it follows that This implies that Subtracting the nth row from the (n + 1)th row, then the (n − 1)th row from the nth row, ..., then the 1st row from the 2nd row of the above determinant leads to which can be readily rearranged as the formula (5). The proof of Theorem 1 is complete.
Remark 1 On 10 May 2016, Dr Wiwat Wanicharpichat at Naresuan University in Thailand told the first author that the matrix is known as the "population projection matrix". See (Kirkland & Neumann, 2013, p. 48, Equation (4.1)).
Remark 2 In the paper (Qi, 2016), an alternative proof of Theorem 1 was given.