# Eulerian number

In combinatorics, the Eulerian number ${\textstyle A(n,k)}$ is the number of permutations of the numbers 1 to ${\textstyle n}$ in which exactly ${\textstyle k}$ elements are greater than the previous element (permutations with ${\textstyle k}$ "ascents").

Leonhard Euler investigated them and associated polynomials in his 1755 book Institutiones calculi differentialis.

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

## Definition

The Eulerian polynomials are defined by the exponential generating function

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

The Eulerian numbers may be defined as the coefficients of the Eulerian polynomials:

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

An explicit formula for ${\textstyle A(n,k)}$ is[1]

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

## Basic properties

• For fixed ${\textstyle n}$ there is a single permutation which has 0 ascents: ${\textstyle (n,n-1,n-2,\ldots ,1)}$. Indeed, as ${\displaystyle {\tbinom {n}{0}}=1}$ for all ${\displaystyle n}$, ${\textstyle A(n,0)=1}$. This formally includes the empty collection of numbers, ${\textstyle n=0}$. And so ${\textstyle A_{0}(t)=A_{1}(t)=1}$.
• For ${\textstyle k=1}$ the explicit formula implies ${\textstyle A(n,1)=2^{n}-(n+1)}$, a sequence in ${\displaystyle n}$ that reads ${\textstyle 0,0,1,4,11,26,57,\dots }$.
• Fully reversing a permutation with ${\textstyle k}$ ascents creates another permutation in which there are ${\textstyle n-k-1}$ ascents. Therefore ${\textstyle A(n,k)=A(n,n-k-1)}$. So there is also a single permutation which has ${\textstyle n-1}$ ascents, namely the rising permutation ${\textstyle (1,2,\ldots ,n)}$. So also ${\textstyle A(n,n-1)}$ equals ${\displaystyle 1}$.
• A lavish upper bound is ${\textstyle A(n,k)\leq (k+1)^{n}\cdot (n+2)^{k}}$. Between the bounds just discussed, the value exceeds ${\displaystyle 1}$.
• For ${\textstyle k\geq n>0}$, the values are formally zero, meaning many sums over ${\textstyle k}$ can be written with an upper index only up to ${\textstyle n-1}$. It also means that the polynomials ${\displaystyle A_{n}(t)}$ are really of degree ${\textstyle n-1}$ for ${\textstyle n>0}$.

A tabulation of the numbers in a triangular array is called the Euler triangle or Euler's triangle. It shares some common characteristics with Pascal's triangle. Values of ${\textstyle A(n,k)}$ (sequence A008292 in the OEIS) for ${\textstyle 0\leq n\leq 9}$ are:

k
n
0 1 2 3 4 5 6 7 8
0 1
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

## Computation

For larger values of ${\textstyle n}$, ${\textstyle A(n,k)}$ can also be calculated using the recursive formula

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

This formula can be motivated from the combinatorial definition and thus serves as a natural starting point for the theory.

For small values of ${\textstyle n}$ and ${\textstyle k}$, the values of ${\textstyle A(n,k)}$ can be calculated by hand. For example

n k Permutations A(n, k)
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) and (3, 1, 2) A(3,1) = 4
2 (1, 2, 3) A(3,2) = 1

Applying the recurrence to one example, we may find

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

Likewise, the Eulerian polynomials can be computed by the recurrence

${\displaystyle A_{0}(t)=1,}$
${\displaystyle A_{n}(t)=A_{n-1}'(t)\cdot t\,(1-t)+A_{n-1}(t)\cdot (1+(n-1)\,t),{\text{ for }}n>1.}$

The second formula can be cast into an inductive form,

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

The following is a python implementation.

import math  # python 3.8

def Ank(n, k) -> int:
"""
Compute A(n, k) using the explicit formula.
"""
def summand(i):
return (-1) ** i * math.comb(n + 1, i) * (k + 1 - i) ** n
return sum(map(summand, range(k + 1)))

def Anks(n) -> list:
"""
Coefficient list for the n'th polynomial A_n(t).
"""
return [1] if n == 0 else [Ank(n, k) for k in range(n)]

def eval_polynomial(coeffs, t):
"""
Polynomial evaluation function.
"""
return sum(c * t ** k for k, c in enumerate(coeffs))

def An(n, t: float) -> float:
"""
Polynomial A_n(t).
"""
return eval_polynomial(Anks(n), t)

# Print the first few polynomials
sup = lambda n: str(n).translate(str.maketrans("0123456789", "⁰¹²³⁴⁵⁶⁷⁸⁹"))
sub = lambda n: str(n).translate(str.maketrans("0123456789", "₀₁₂₃₄₅₆₇₈₉"))

NUM = 8
for n in range(NUM):
print(f"A{sub(n)}(t) = " + " + ".join(f"{ank} t{sup(k)}" for k, ank in enumerate(Anks(n))))
# E.g. A₇(t) = 1 t⁰ + 120 t¹ + 1191 t² + 2416 t³ + 1191 t⁴ + 120 t⁵ + 1 t⁶

# Sanity checks
assert Ank(n, 1) == 2 ** n - (n + 1)
assert n == 0 or An(n, 1) == math.factorial(n)  # Cardinality check
alternating_sum: float = sum((-1)**k * Ank(n, k) / math.comb(n - 1, k) for k in range(n))
assert n < 2 or abs(alternating_sum) < 1e-13


## Identities

For any property partitioning a finite set into finitely many smaller sets, the sum of the cardinalities of the smaller sets equals the cardinality of the bigger set. The Eulerian numbers partition the permutations of ${\displaystyle n}$ elements, so their sum equals the factorial ${\displaystyle n!}$. I.e.

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

as well as ${\displaystyle A(0,0)=0!}$. To avoid conflict with the empty sum convention, it is convenient to simply state the theorems for ${\displaystyle n>0}$ only.

Much more generally, for a fixed function ${\displaystyle f\colon \mathbb {R} \rightarrow \mathbb {C} }$ integrable on the interval ${\displaystyle (0,n)}$[2]

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

Worpitzky's identity[3] expresses ${\textstyle x^{n}}$ as the linear combination of Eulerian numbers with binomial coefficients:

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

From it, it follows that

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

### Formulas involving alternating sums

The alternating sum of the Eulerian numbers for a fixed value of ${\textstyle n}$ is related to the Bernoulli number ${\textstyle B_{n+1}}$

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

Furthermore,

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

and

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

### Formulas involving the polynomials

The symmetry property implies:

${\displaystyle A_{n}(t)=t^{n-1}A_{n}(t^{-1})}$

The Eulerian numbers are involved in the generating function for the sequence of nth powers:

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

It also holds that,

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

## Eulerian numbers of the second order

The permutations of the multiset ${\textstyle \{1,1,2,2,\ldots ,n,n\}}$ 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 ${\textstyle (2n-1)!!}$. 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:

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

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

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

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

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

and the above recurrence relations are translated into a recurrence relation for the sequence Pn(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 somewhat 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 ${\textstyle P_{n}(x)=(1-x)^{2n}u_{n}(x)}$, and the Eulerian numbers of second order as their coefficients.

The following table displays the first few second-order Eulerian numbers:

k
n
0 1 2 3 4 5 6 7 8
0 1
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 ${\textstyle P_{n}(1)}$, is ${\textstyle (2n-1)!!}$.

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,
• (sequence A340556 in the OEIS), extending the definition of Gessel and Stanley.

## References

• Eulerus, Leonardus [Leonhard Euler] (1755). Institutiones calculi differentialis cum eius usu in analysi finitorum ac doctrina serierum [Foundations of differential calculus, with applications to finite analysis and series]. Academia imperialis scientiarum Petropolitana; Berolini: Officina Michaelis.
• Carlitz, L. (1959). "Eulerian Numbers and polynomials". Math. Mag. 32 (5): 247–260. doi:10.2307/3029225. JSTOR 3029225.
• Gould, H. W. (1978). "Evaluation of sums of convolved powers using Stirling and Eulerian Numbers". Fib. Quart. 16 (6): 488–497.
• Desarmenien, Jacques; Foata, Dominique (1992). "The signed Eulerian numbers". Discrete Math. 99 (1–3): 49–58. doi:10.1016/0012-365X(92)90364-L.
• Lesieur, Leonce; Nicolas, Jean-Louis (1992). "On the Eulerian Numbers M=max (A(n,k))". Europ. J. Combinat. 13 (5): 379–399. doi:10.1016/S0195-6698(05)80018-6.
• Butzer, P. L.; Hauss, M. (1993). "Eulerian numbers with fractional order parameters". Aequationes Mathematicae. 46 (1–2): 119–142. doi:10.1007/bf01834003. S2CID 121868847.
• Koutras, M.V. (1994). "Eulerian numbers associated with sequences of polynomials". Fib. Quart. 32 (1): 44.
• Graham; Knuth; Patashnik (1994). Concrete Mathematics: A Foundation for Computer Science (2nd ed.). Addison-Wesley. pp. 267–272.
• Hsu, Leetsch C.; Jau-Shyong Shiue, Peter (1999). "On certain summation problems and generalizations of Eulerian polynomials and numbers". Discrete Math. 204 (1–3): 237–247. doi:10.1016/S0012-365X(98)00379-3.
• Boyadzhiev, Khristo N. (2007). "Apostol-Bernoulli functions, derivative polynomials and Eulerian polynomials". arXiv:0710.1124 [math.CA].
• Petersen, T. Kyle (2015). Eulerian Numbers. Birkhäuser Advanced Texts Basler Lehrbücher. Birkhäuser. pp. 3–18. doi:10.1007/978-1-4939-3091-3_1. ISBN 978-1-4939-3090-6.

## Citations

1. ^ (L. Comtet 1974, p. 243)
2. ^ Exercise 6.65 in Concrete Mathematics by Graham, Knuth and Patashnik.
3. ^ Worpitzky, J. (1883). "Studien über die Bernoullischen und Eulerschen Zahlen". Journal für die reine und angewandte Mathematik. 94: 203–232.