In combinatorics, the Eulerian number A(n, m) is the number of permutations of the numbers 1 to n in which exactly m elements are greater than the previous element (permutations with m "ascents"). They are the coefficients of the Eulerian polynomials:

${\displaystyle A_{n}(t)=\sum _{m=0}^{n}A(n,m)\ t^{m}.}$

The Eulerian polynomials are defined by the exponential generating function

${\displaystyle \sum _{n=0}^{\infty }A_{n}(t)\cdot {\frac {x^{n)){n!))={\frac {t-1}{t-\mathrm {e} ^{(t-1)x))}.}$

The Eulerian polynomials can be computed by the recurrence

${\displaystyle A_{0}(t)=1,}$

${\displaystyle A_{n}(t)=t(1-t)\cdot A_{n-1}'(t)+A_{n-1}(t)\cdot (1+(n-1)t),\quad n\geq 1.}$

An equivalent way to write this definition is to set the Eulerian polynomials inductively by

${\displaystyle A_{0}(t)=1,}$

${\displaystyle A_{n}(t)=\sum _{k=0}^{n-1}{\binom {n}{k))A_{k}(t)\cdot (t-1)^{n-1-k},\quad n\geq 1.}$

Other notations for A(n, m) are E(n, m) and ${\displaystyle \textstyle \left\langle {n \atop m}\right\rangle }$.

History

Shows the polynomials presently known as Eulerian polynomials in Euler's work from 1755, Institutiones calculi differentialis, part 2, p. 485/6. The coefficients of these polynomials are known as Eulerian numbers.

In 1755, in his book Institutiones calculi differentialis, Leonhard Euler investigated polynomials α₁(x) = 1, α₂(x) = x + 1, α₃(x) = x² + 4x + 1, etc. (see the facsimile). These polynomials are a shifted form of what are now called the Eulerian polynomials A_n(x).

Basic properties

For a given value of n > 0, the index m in A(n, m) can take values from 0 to n − 1. For fixed n there is a single permutation which has 0 ascents: (n, n − 1, n − 2, ..., 1). There is also a single permutation which has n − 1 ascents; this is the rising permutation (1, 2, 3, ..., n). Therefore A(n, 0) and A(n, n − 1) 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(n, m) = A(n, n − m − 1).

Values of A(n, m) can be calculated "by hand" for small values of n and m. For example

n	m	Permutations	A(n, m)
1	0	(1)	A(1,0) = 1
2	0	(2, 1)	A(2,0) = 1
	1	(1, 2)	A(2,1) = 1
3	0	(3, 2, 1)	A(3,0) = 1
	1	(1, 3, 2) (2, 1, 3) (2, 3, 1) (3, 1, 2)	A(3,1) = 4
	2	(1, 2, 3)	A(3,2) = 1

For larger values of n, A(n, m) can be calculated using the recursive formula

${\displaystyle A(n,m)=(n-m)A(n-1,m-1)+(m+1)A(n-1,m).}$

For example

${\displaystyle A(4,1)=(4-1)A(3,0)+(1+1)A(3,1)=3\times 1+2\times 4=11.}$

Values of A(n, m) (sequence A008292 in the OEIS) for 0 ≤ n ≤ 9 are:

m n	0	1	2	3	4	5	6	7	8
1	1
2	1	1
3	1	4	1
4	1	11	11	1
5	1	26	66	26	1
6	1	57	302	302	57	1
7	1	120	1191	2416	1191	120	1
8	1	247	4293	15619	15619	4293	247	1
9	1	502	14608	88234	156190	88234	14608	502	1

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(n, m) is^[1]

${\displaystyle A(n,m)=\sum _{k=0}^{m+1}(-1)^{k}{\binom {n+1}{k))(m+1-k)^{n}.}$

One can see from this formula, as well as from the combinatorial interpretation, that ${\displaystyle A(n,n)=0}$ for ${\displaystyle n>0}$, so that ${\displaystyle A_{n}(t)}$ is a polynomial of degree ${\displaystyle n-1}$ for ${\displaystyle n>0}$.

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

${\displaystyle \sum _{m=0}^{n-1}A(n,m)=n!{\text{ for ))n\geq 1.}$

The alternating sum of the Eulerian numbers for a fixed value of n is related to the Bernoulli number B_n+1

${\displaystyle \sum _{m=0}^{n-1}(-1)^{m}A(n,m)={\frac {2^{n+1}(2^{n+1}-1)B_{n+1)){n+1)){\text{ for ))n\geq 1.}$

Other summation properties of the Eulerian numbers are:

${\displaystyle \sum _{m=0}^{n-1}(-1)^{m}{\frac {A(n,m)}{\binom {n-1}{m))}=0{\text{ for ))n\geq 2,}$

${\displaystyle \sum _{m=0}^{n-1}(-1)^{m}{\frac {A(n,m)}{\binom {n}{m))}=(n+1)B_{n}{\text{ for ))n\geq 2,}$

where B_n is the n^th Bernoulli number.

Identities

The Eulerian numbers are involved in the generating function for the sequence of n^th powers:

${\displaystyle \sum _{k=0}^{\infty }k^{n}x^{k}={\frac {\sum _{m=0}^{n-1}A(n,m)x^{m+1)){(1-x)^{n+1))}={\frac {x\cdot A_{n}(x)}{(1-x)^{n+1))))$

for ${\displaystyle n\geq 0}$. This assumes that 0⁰ = 0 and A(0,0) = 1 (since there is one permutation of no elements, and it has no ascents).

Worpitzky's identity^[2] expresses xⁿ as the linear combination of Eulerian numbers with binomial coefficients:

${\displaystyle x^{n}=\sum _{m=0}^{n-1}A(n,m){\binom {x+m}{n)).}$

It follows from Worpitzky's identity that

${\displaystyle \sum _{k=1}^{N}k^{n}=\sum _{m=0}^{n-1}A(n,m){\binom {N+1+m}{n+1)).}$

Another interesting identity is

${\displaystyle {\frac {e}{1-ex))=\sum _{n=0}^{\infty }{\frac {A_{n}(x)}{n!(1-x)^{n+1))}.}$

The numerator on the right-hand side is the Eulerian polynomial.

For a fixed function ${\displaystyle f:\mathbb {R} \rightarrow \mathbb {C} }$ which is integrable on ${\displaystyle (0,n)}$ we have the integral formula ^[3]

${\displaystyle \int _{0}^{1}\cdots \int _{0}^{1}f\left(\left\lfloor x_{1}+\cdots +x_{n}\right\rfloor \right)dx_{1}\cdots dx_{n}=\sum _{k}A(n,k){\frac {f(k)}{n!)).}$

Eulerian numbers of the second order

The permutations of the multiset {1, 1, 2, 2, ···, n, n} 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 (2n−1)!!. The Eulerian number of the second order, denoted ${\displaystyle \scriptstyle \left\langle \!\left\langle {n \atop m}\right\rangle \!\right\rangle }$, 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:

332211,

221133, 221331, 223311, 233211, 113322, 133221, 331122, 331221,

112233, 122133, 112332, 123321, 133122, 122331.

The Eulerian numbers of the second order satisfy the recurrence relation, that follows directly from the above definition:

${\displaystyle \left\langle \!\!\left\langle {n \atop m}\right\rangle \!\!\right\rangle =(2n-m-1)\left\langle \!\!\left\langle {n-1 \atop m-1}\right\rangle \!\!\right\rangle +(m+1)\left\langle \!\!\left\langle {n-1 \atop m}\right\rangle \!\!\right\rangle ,}$

with initial condition for n = 0, expressed in Iverson bracket notation:

${\displaystyle \left\langle \!\!\left\langle {0 \atop m}\right\rangle \!\!\right\rangle =[m=0].}$

Correspondingly, the Eulerian polynomial of second order, here denoted P_n (no standard notation exists for them) are

${\displaystyle P_{n}(x):=\sum _{m=0}^{n}\left\langle \!\!\left\langle {n \atop m}\right\rangle \!\!\right\rangle x^{m))$

and the above recurrence relations are translated into a recurrence relation for the sequence P_n(x):

${\displaystyle P_{n+1}(x)=(2nx+1)P_{n}(x)-x(x-1)P_{n}^{\prime }(x)}$

with initial condition

${\displaystyle P_{0}(x)=1.}$

The latter recurrence may be written in a somehow more compact form by means of an integrating factor:

${\displaystyle (x-1)^{-2n-2}P_{n+1}(x)=\left(x(1-x)^{-2n-1}P_{n}(x)\right)^{\prime ))$

so that the rational function

${\displaystyle u_{n}(x):=(x-1)^{-2n}P_{n}(x)}$

satisfies a simple autonomous recurrence:

${\displaystyle u_{n+1}=\left({\frac {x}{1-x))u_{n}\right)^{\prime },\quad u_{0}=1,}$

whence one obtains the Eulerian polynomials of second order as P_n(x) = (1−x)²ⁿ u_n(x), and the Eulerian numbers of second order as their coefficients.

Here are some values of the second order Eulerian numbers:

m n	0	1	2	3	4	5	6	7	8
1	1
2	1	2
3	1	8	6
4	1	22	58	24
5	1	52	328	444	120
6	1	114	1452	4400	3708	720
7	1	240	5610	32120	58140	33984	5040
8	1	494	19950	195800	644020	785304	341136	40320
9	1	1004	67260	1062500	5765500	12440064	11026296	3733920	362880

The sum of the n-th row, which is also the value P_n(1), is (2n − 1)!!.

Indexing the second-order Eulerian numbers comes in three flavors: (sequence A008517 in the OEIS) following Riordan and Comtet, (sequence A201637 in the OEIS) following Graham, Knuth, and Patashnik and (sequence A340556 in the OEIS), extending the definition of Gessel and Stanley.