Eulerian number


In combinatorics, the Eulerian number A is the number of permutations of the numbers 1 to n in which exactly m elements are greater than the previous element. They are the coefficients of the Eulerian polynomials:
The Eulerian polynomials are defined by the exponential generating function
The Eulerian polynomials can be computed by the recurrence
An equivalent way to write this definition is to set the Eulerian polynomials inductively by
Other notations for A are E and.

History

In 1755 Leonhard Euler investigated in his book Institutiones calculi differentialis polynomials,,, etc.. These polynomials are a shifted form of what are now called the Eulerian polynomials An.

Basic properties

For a given value of n > 0, the index m in A can take values from 0 to n − 1. For fixed n there is a single permutation which has 0 ascents:. There is also a single permutation which has n − 1 ascents; this is the rising permutation. Therefore A and A are 1 for all values of n.
Reversing a permutation with m ascents creates another permutation in which there are n − m − 1 ascents.
Therefore A = A.
Values of A can be calculated "by hand" for small values of n and m. For example
For larger values of n, A can be calculated using the recursive formula
For example
Values of A for 0 ≤ n ≤ 9 are:
The above triangular array is called the Euler triangle or Euler's triangle, and it shares some common characteristics with Pascal's triangle. The sum of row n is the factorial n!.

Explicit formula

An explicit formula for A is
One can see from this formula, as well as from the combinatorial interpretation, that for, so that is a polynomial of degree for.

Summation properties

It is clear from the combinatorial definition that the sum of the Eulerian numbers for a fixed value of n is the total number of permutations of the numbers 1 to n, so
The alternating sum of the Eulerian numbers for a fixed value of n is related to the Bernoulli number Bn+1
Other summation properties of the Eulerian numbers are:
where Bn is the nth Bernoulli number.

Identities

The Eulerian numbers are involved in the generating function for the sequence of nth powers:
for. This assumes that 00 = 0 and A = 1.
Worpitzky's identity expresses xn as the linear combination of Eulerian numbers with binomial coefficients:
It follows from Worpitzky's identity that
Another interesting identity is
The numerator on the right-hand side is the Eulerian polynomial.
For a fixed function which is integrable on we have the integral formula

Eulerian numbers of the second kind

The permutations of the multiset which have the property that for each k, all the numbers appearing between the two occurrences of k in the permutation are greater than k are counted by the double factorial number !!.
The Eulerian number of the second kind, denoted, counts the number of all such permutations that have exactly m ascents. For instance, for n = 3 there are 15 such permutations, 1 with no ascents, 8 with a single ascent, and 6 with two ascents:
The Eulerian numbers of the second kind satisfy the recurrence relation, that follows directly from the above definition:
with initial condition for n = 0, expressed in Iverson bracket notation:
Correspondingly, the Eulerian polynomial of second kind, here denoted Pn are
and the above recurrence relations are translated into a recurrence relation for the sequence Pn:
with initial condition
The latter recurrence may be written in a somehow more compact form by means of an integrating factor:
so that the rational function
satisfies a simple autonomous recurrence:
whence one obtains the Eulerian polynomials as Pn = 2n un, and the Eulerian numbers of the second kind as their coefficients.
Here are some values of the second order Eulerian numbers :
The sum of the n-th row, which is also the value Pn, is !!.

Citations