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Definition

The Eulerian polynomials A n ( t ) {\displaystyle A_{n}(t)} are defined by the exponential generating function

∑ n = 0 ∞ A n ( t ) x n n ! = t − 1 t − e ( t − 1 ) x = ( 1 − e ( t − 1 ) x − 1 t − 1 ) − 1 . {\displaystyle \sum _{n=0}^{\infty }A_{n}(t)\,{\frac {x^{n}}{n!}}={\frac {t-1}{t-e^{(t-1)\,x}}}=\left(1-{\frac {e^{(t-1)x}-1}{t-1}}\right)^{-1}.}

The Eulerian numbers A ( n , k ) {\displaystyle A(n,k)} may be defined as the coefficients of the Eulerian polynomials:

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

An explicit formula for A ( n , k ) {\textstyle A(n,k)} is¹

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

Basic properties

• For fixed n {\textstyle n} there is a single permutation which has 0 ascents: ( n , n − 1 , n − 2 , … , 1 ) {\textstyle (n,n-1,n-2,\ldots ,1)} . Indeed, as ( n 0 ) = 1 {\displaystyle {\tbinom {n}{0}}=1} for all n {\displaystyle n} , A ( n , 0 ) = 1 {\textstyle A(n,0)=1} . This formally includes the empty collection of numbers, n = 0 {\textstyle n=0} . And so A 0 ( t ) = A 1 ( t ) = 1 {\textstyle A_{0}(t)=A_{1}(t)=1} .
• For k = 1 {\textstyle k=1} the explicit formula implies A ( n , 1 ) = 2 n − ( n + 1 ) {\textstyle A(n,1)=2^{n}-(n+1)} , a sequence in n {\displaystyle n} that reads 0 , 0 , 1 , 4 , 11 , 26 , 57 , … {\textstyle 0,0,1,4,11,26,57,\dots } .
• Fully reversing a permutation with k {\textstyle k} ascents creates another permutation in which there are n − k − 1 {\textstyle n-k-1} ascents. Therefore A ( n , k ) = A ( n , n − k − 1 ) {\textstyle A(n,k)=A(n,n-k-1)} . So there is also a single permutation which has n − 1 {\textstyle n-1} ascents, namely the rising permutation ( 1 , 2 , … , n ) {\textstyle (1,2,\ldots ,n)} . So also A ( n , n − 1 ) {\textstyle A(n,n-1)} equals 1 {\displaystyle 1} .
• A lavish upper bound is A ( n , k ) ≤ ( k + 1 ) n ⋅ ( n + 2 ) k {\textstyle A(n,k)\leq (k+1)^{n}\cdot (n+2)^{k}} . Between the bounds just discussed, the value exceeds 1 {\displaystyle 1} .
• For k ≥ n > 0 {\textstyle k\geq n>0} , the values are formally zero, meaning many sums over k {\textstyle k} can be written with an upper index only up to n − 1 {\textstyle n-1} . It also means that the polynomials A n ( t ) {\displaystyle A_{n}(t)} are really of degree n − 1 {\textstyle n-1} for n > 0 {\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 A ( n , k ) {\textstyle A(n,k)} (sequence A008292 in the OEIS) for 0 ≤ n ≤ 9 {\textstyle 0\leq n\leq 9} are:

kn	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 n {\textstyle n} , A ( n , k ) {\textstyle A(n,k)} can also be calculated using the recursive formula²

A ( n , k ) = ( n − k ) A ( n − 1 , k − 1 ) + ( k + 1 ) A ( n − 1 , k ) . {\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 n {\textstyle n} and k {\textstyle k} , the values of A ( n , k ) {\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

A ( 4 , 1 ) = ( 4 − 1 ) A ( 3 , 0 ) + ( 1 + 1 ) A ( 3 , 1 ) = 3 ⋅ 1 + 2 ⋅ 4 = 11. {\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

A 0 ( t ) = 1 , {\displaystyle A_{0}(t)=1,} A n ( t ) = A n − 1 ′ ( t ) ⋅ t ( 1 − t ) + A n − 1 ( t ) ⋅ ( 1 + ( n − 1 ) t ) ,  for  n > 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,

A n ( t ) = ∑ k = 0 n − 1 ( n k ) A k ( t ) ⋅ ( t − 1 ) n − 1 − k ,  for  n > 1. {\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.}

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 n {\displaystyle n} elements, so their sum equals the factorial n ! {\displaystyle n!} . I.e.

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

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

Much more generally, for a fixed function f : R → C {\displaystyle f\colon \mathbb {R} \rightarrow \mathbb {C} } integrable on the interval ( 0 , n ) {\displaystyle (0,n)} ³

∑ k = 0 n − 1 A ( n , k ) f ( k ) = n ! ∫ 0 1 ⋯ ∫ 0 1 f ( ⌊ x 1 + ⋯ + x n ⌋ ) d x 1 ⋯ d x n {\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 identity⁴ expresses x n {\textstyle x^{n}} as the linear combination of Eulerian numbers with binomial coefficients:

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

From it, it follows that

∑ k = 1 m k n = ∑ k = 0 n − 1 A ( n , k ) ( m + k + 1 n + 1 ) . {\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 n {\textstyle n} is related to the Bernoulli number B n + 1 {\textstyle B_{n+1}}

∑ k = 0 n − 1 ( − 1 ) k A ( n , k ) = 2 n + 1 ( 2 n + 1 − 1 ) B n + 1 n + 1 ,  for  n > 0. {\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,

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

and

∑ k = 0 n − 1 ( − 1 ) k A ( n , k ) ( n k ) = ( n + 1 ) B n ,  for  n > 1 {\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:

A n ( t ) = t n − 1 A n ( t − 1 ) {\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:

∑ i = 1 ∞ i n x i = 1 ( 1 − x ) n + 1 ∑ k = 0 n A ( n , k ) x k + 1 = x ( 1 − x ) n + 1 A n ( x ) {\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)}

An explicit expression for Eulerian polynomials is⁵

A n ( t ) = ∑ k = 0 n { n k } k ! ( t − 1 ) n − k {\displaystyle A_{n}(t)=\sum _{k=0}^{n}\left\{{n \atop k}\right\}k!(t-1)^{n-k}}

where { n k } {\textstyle \left\{{n \atop k}\right\}} is the Stirling number of the second kind.

Eulerian numbers of the second order

The permutations of the multiset { 1 , 1 , 2 , 2 , … , n , n } {\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 ( 2 n − 1 ) ! ! {\textstyle (2n-1)!!} . The Eulerian number of the second order, denoted ⟨ ⟨ n m ⟩ ⟩ {\textstyle \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:

⟨ ⟨ n k ⟩ ⟩ = ( 2 n − k − 1 ) ⟨ ⟨ n − 1 k − 1 ⟩ ⟩ + ( k + 1 ) ⟨ ⟨ n − 1 k ⟩ ⟩ , {\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:

⟨ ⟨ 0 k ⟩ ⟩ = [ k = 0 ] . {\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

P n ( x ) := ∑ k = 0 n ⟨ ⟨ n k ⟩ ⟩ x k {\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):

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

with initial condition P 0 ( x ) = 1. {\displaystyle P_{0}(x)=1.} . The latter recurrence may be written in a somewhat more compact form by means of an integrating factor:

( x − 1 ) − 2 n − 2 P n + 1 ( x ) = ( x ( 1 − x ) − 2 n − 1 P n ( x ) ) ′ {\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

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

satisfies a simple autonomous recurrence:

u n + 1 = ( x 1 − x u n ) ′ , u 0 = 1 {\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 ) 2 n u n ( x ) {\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:

kn	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 P n ( 1 ) {\textstyle P_{n}(1)} , is ( 2 n − 1 ) ! ! {\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.

• 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. doi:10.1080/00150517.1978.12430271.
• 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): 44–57. doi:10.1080/00150517.1994.12429255.
• 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". 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

External links

• Eulerian Polynomials at OEIS Wiki.
• "Eulerian Numbers". MathPages.com.
• Weisstein, Eric W. "Eulerian Number". MathWorld.
• Weisstein, Eric W. "Euler's Number Triangle". MathWorld.
• Weisstein, Eric W. "Worpitzky's Identity". MathWorld.
• Weisstein, Eric W. "Second-Order Eulerian Triangle". MathWorld.
• Euler-matrix (generalized rowindexes, divergent summation)

References

1. (L. Comtet 1974, p. 243) ↩

2. Comtet, Louis. Advanced Combinatorics (PDF). p. 51. https://ia801306.us.archive.org/27/items/Comtet_Louis_-_Advanced_Coatorics/Comtet_Louis_-_Advanced_Combinatorics.pdf ↩

3. Exercise 6.65 in Concrete Mathematics by Graham, Knuth and Patashnik. ↩

4. Worpitzky, J. (1883). "Studien über die Bernoullischen und Eulerschen Zahlen". Journal für die reine und angewandte Mathematik. 94: 203–232. https://eudml.org/doc/148532 ↩

5. Qi, Feng; Guo, Bai-Ni (2017-08-01). "Explicit formulas and recurrence relations for higher order Eulerian polynomials". Indagationes Mathematicae. 28 (4): 884–891. doi:10.1016/j.indag.2017.06.010. ISSN 0019-3577. https://doi.org/10.1016%2Fj.indag.2017.06.010 ↩