Eulerian number

{{Short description|Polynomial sequence}} {{distinguish|Euler number|Euler's number}} {{Use American English|date = March 2019}} In combinatorics, the '''Eulerian number''' <math display="inline">A(n,k)</math> is the number of permutations of the numbers 1 to ''<math display="inline">n</math>'' in which exactly ''<math display="inline">k</math>'' elements are greater than the previous element (permutations with ''<math display="inline">k</math>'' "ascents").

Leonhard Euler investigated them and associated polynomials in his 1755 book ''Institutiones calculi differentialis''.<ref>{{Cite journal |last=Euler |first=Leonhard |date=1755-01-01 |title=Institutiones calculi differentialis cum eius usu in analysi finitorum ac doctrina serierum, volume 1 |url=https://scholarlycommons.pacific.edu/euler-works/212 |journal=Academiae Imperialis Scientiarum Petropolitanae |pages=1–880}}</ref><ref>{{Citation |last=Euler |first=Leonhard |title=Institutiones calculi differentialis cum eius usu in analysi finitorum ac doctrina serierum |date=2019-05-24 |url=http://arxiv.org/abs/1905.10438 |access-date=2026-04-21 |publisher=arXiv |doi=10.48550/arXiv.1905.10438 |id=arXiv:1905.10438 |last2=Aycock |first2=Alexander}}</ref>

right|303px|thumb|The polynomials presently known as Eulerian polynomials in Euler's work from 1755, Institutiones calculi differentialis, part 2, §173, p. 485/6. The coefficients of these polynomials are known as Eulerian numbers.

Other notations for <math display="inline">A(n,k)</math> are <math display="inline">E(n,k)</math> and <math>\textstyle \left\langle {n \atop k} \right\rangle</math>.

==Definition== alt=A plot of the Eulerian numbers with the second argument fixed to 5.|thumb|A plot of the Eulerian numbers with the second argument fixed to 5.

The '''Eulerian polynomials''' <math>A_n(t)</math> are defined by the exponential generating function :<math>\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}.</math> The '''Eulerian numbers''' <math>A(n,k)</math> may also be defined as the coefficients of the Eulerian polynomials: :<math>A_{n}(t) = \sum_{k=0}^n A(n,k)\,t^k.</math>

An explicit formula for <math display="inline">A(n,k)</math> is{{sfnp|Comtet|1974|p=243}} :<math>A(n,k)=\sum_{i=0}^{k}(-1)^i \binom{n+1}{i} (k+1-i)^n.</math>

==Basic properties==<!--linked to by Euler triangle & Euler's triangle--> * For fixed ''<math display="inline">n</math>'' there is a single permutation which has 0 ascents: <math display="inline">(n, n-1, n-2, \ldots, 1)</math>. Indeed, as <math>{\tbinom n 0}=1</math> for all <math>n</math>, <math display="inline">A(n, 0) = 1</math>. This formally includes the empty collection of numbers, <math display="inline">n=0</math>. And so <math display="inline">A_0(t)=A_1(t)=1</math>. * For <math display="inline">k=1</math> the explicit formula implies <math display="inline">A(n,1)=2^n-(n+1)</math>, a sequence in <math>n</math> that reads <math display="inline">0, 0, 1, 4, 11, 26, 57, \dots</math>. * Fully reversing a permutation with ''<math display="inline">k</math>'' ascents creates another permutation in which there are ''<math display="inline">n-k-1</math>'' ascents. Therefore <math display="inline">A(n, k) = A(n, n-k-1)</math>. So there is also a single permutation which has ''<math display="inline">n-1</math>'' ascents, namely the rising permutation <math display="inline">(1, 2, \ldots, n)</math>. So also <math display="inline">A(n, n-1)</math> equals <math>1</math>. * Since a permutation of the numbers <math>1</math> to <math>n</math> which has <math>k</math> ascents must have <math>n-1-k</math> descents, the symmetry <math display="inline">A(n, k) = A(n, n-k-1)</math> shows that <math display="inline">A(n, k)</math> also counts the number of permutations with <math>k</math> ''descents''. * For <math display="inline">k\ge n > 0</math>, the values are formally zero, meaning many sums over <math display="inline">k</math> can be written with an upper index only up to <math display="inline">n-1</math>. It also means that the polynomials <math>A_{n}(t)</math> are really of degree <math display="inline">n-1</math> for <math display="inline">n>0</math>.

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 <math display="inline">A(n, k)</math> {{OEIS|id=A008292}} for <math display="inline">0 \le n \le 9</math> are:

:{| class="wikitable" style="text-align:right;" |- ! {{diagonal split header|''n''&nbsp;|&nbsp;''k''}} ! width="50" | 0 ! width="50" | 1 ! width="50" | 2 ! width="50" | 3 ! width="50" | 4 ! width="50" | 5 ! width="50" | 6 ! width="50" | 7 ! width="50" | 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 <math display="inline">n</math>, <math display="inline">A(n,k)</math> can also be calculated using the recursive formula{{sfnp|Comtet|1974|p=51}} :<math>A(n,k)=(n-k)A(n-1,k-1) + (k+1)A(n-1,k).</math> This formula can be motivated from the combinatorial definition and thus serves as a natural starting point for the theory.

For small values of <math display="inline">n</math> and <math display="inline">k</math>, the values of <math display="inline">A(n,k)</math> can be calculated by hand. For example

:{| class="wikitable" |- ! ''n'' ! ''k'' ! Permutations ! ''A''(''n'', ''k'') |- | 1 | 0 | (1) | ''A''(1,0) = 1 |- |rowspan="2" | 2 | 0 | (2, 1) | ''A''(2,0) = 1 |- | 1 | (1, '''2''') | ''A''(2,1) = 1 |- |rowspan="3" | 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 :<math>A(4,1)=(4-1)\,A(3,0) + (1+1)\,A(3,1)=3 \cdot 1 + 2 \cdot 4 = 11.</math>

Likewise, the Eulerian polynomials can be computed by the recurrence :<math>A_{0}(t) = 1,</math> :<math>A_{n}(t) = A_{n-1}'(t)\cdot t\,(1-t) + A_{n-1}(t)\cdot (1+(n-1)\,t),\text{ for } n > 1.</math> The second formula can be cast into an inductive form, :<math>A_{n}(t) = \sum_{k=0}^{n-1} \binom{n}{k} A_{k}(t)\cdot (t-1)^{n-1-k}, \text{ for } n > 1.</math>

==Identities== For any set partition of a finite set into disjoint subsets, the sum of the cardinalities of the parts equals the cardinality of the whole set. Since there are <math>n!</math> (the factorial of <math>n</math>) permutations of size <math>n</math>, and the Eulerian numbers give the cardinalities of the subsets of these permutations with fixed number of descents, it follows that <math display="block">\sum_{k=0}^{n} A(n,k) = n! \,.</math> (The summand <math>k = n</math> is 0 for <math>n>0</math>, but is included to give the correct sum <math>A(0,0)=0!</math> when <math>n = 0</math>.) Much more generally, for a fixed function <math>f\colon \mathbb{R} \rightarrow \mathbb{C}</math> integrable on the interval <math>[0, n]</math>, the following identity holds:{{sfnp|Graham|Knuth|Patashnik|1994|loc=Exercise 6.65}} :<math>\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) dx_1 \cdots dx_n .</math>

'''Worpitzky's identity''' expresses <math display="inline">x^n</math> as the linear combination of Eulerian numbers with binomial coefficients: :<math>\sum_{k=0}^{n-1}A(n,k)\binom{x+k}{n}=x^n.</math> From this, it follows that :<math>\sum_{k=1}^{m}k^n=\sum_{k=0}^{n-1} A(n,k) \binom{m+k+1}{n+1}.</math> This identity is named after Julius Worpitzky, who discovered it in the 1880s;<ref>{{cite journal |last1=Worpitzky |first1=J. |title=Studien über die Bernoullischen und Eulerschen Zahlen |journal=Journal für die reine und angewandte Mathematik |date=1883 |volume=94 |pages=203–232 |url=https://eudml.org/doc/148532}}</ref> it had been originally discovered somewhat earlier, by Li Shanlan in his 1867 work ''Duò Jī Bǐ Lèi''.{{sfnp|Petersen|2015|p=14}}<ref>{{Cite book |last=Knuth |first=Donald Ervin |title=The art of computer programming |date=1997 |publisher=Addison-Wesley |isbn=978-0-201-89683-1 |edition=3rd |location=Reading, Mass |pages=36}}</ref>

Eulerian numbers appear as the coefficients of the polylogarithm for negative integer inputs:<math display="block">\operatorname{Li}_{-n}(z) = {1 \over (1-z)^{n+1}} \sum_{k=0}^{n-1} A(n, k) z^{n-k} \qquad (n=1,2,3,\ldots).</math>

===Formulas involving alternating sums=== The alternating sum of the Eulerian numbers for a fixed value of ''<math display="inline">n</math>'' is related to the Bernoulli number <math display="inline">B_{n+1}</math> :<math>\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.</math> Furthermore, :<math>\sum_{k=0}^{n-1}(-1)^k \frac{A(n,k)}{\binom{n-1}{k}}=0, \text{ for }n > 1</math> and :<math>\sum_{k=0}^{n-1}(-1)^k \frac{A(n,k)}{\binom{n}{k}}=(n+1)B_{n}, \text{ for } n > 1</math>

===Formulas involving the polynomials=== The symmetry property implies: :<math>A_n(t) = t^{n-1} A_n(t^{-1}) </math> The Eulerian numbers are involved in the generating function for the sequence of ''n''^th powers: :<math>\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)</math> An explicit expression for Eulerian polynomials is<ref>{{Cite journal |last1=Qi |first1=Feng |last2=Guo |first2=Bai-Ni |date=2017-08-01 |title=Explicit formulas and recurrence relations for higher order Eulerian polynomials |journal=Indagationes Mathematicae |volume=28 |issue=4 |pages=884–891 |doi=10.1016/j.indag.2017.06.010 |issn=0019-3577|doi-access=free }}</ref>

<math display="block">A_n(t) = \sum_{k=0}^n \left\{ {n \atop k} \right\} k! (t-1)^{n-k}</math>

where <math display="inline">\left\{ {n \atop k} \right\}</math> is the Stirling number of the second kind.

==Geometric interpretations==

The Eulerian numbers have two important geometric interpretations involving convex polytopes.

First of all, the identity :<math>\sum_{i=0}^\infty (i+1)^n x^i = \frac{1}{(1-x)^{n+1}}\sum_{k=0}^n A(n,k)\,x^{k} </math> implies that the Eulerian numbers form the <math>h^\ast</math>-vector of the standard <math>n</math>-dimensional hypercube, which is the convex hull of all <math>0,1</math>-vectors in <math>\mathbb{R}^n</math>.

Secondly, the identity <math display="block">A_n(t) = \sum_{k=0}^n \left\{ {n \atop k} \right\} k! (t-1)^{n-k}</math> means that the Eulerian numbers also form the <math>h</math>-vector of the simple polytope which is dual to the <math>n</math>-dimensional permutohedron, which is the convex hull of all permutations of the vector <math>(1,2,\ldots,n)</math> in <math>\mathbb{R}^n</math>.

==Type B Eulerian numbers==

The hyperoctahedral group of order <math>n</math> is the group of all ''signed permutations'' of the numbers <math>1</math> to <math>n</math>, meaning bijections <math>\pi</math> from the set <math>\{-n,-n+1,\ldots,-1,1,2,\ldots,n\}</math> to itself with the property that <math>\pi(-i)=-\pi(i)</math> for all <math>i</math>. Just as the symmetric group of order <math>n</math> (i.e., the group of all permutations of the numbers <math>1</math> to <math>n</math>) is the Coxeter group of Type <math>A_{n-1}</math>, the hyperoctahedral group of order <math>n</math> is the Coxeter group of Type <math>B_n</math>.

Given an element <math>\pi</math> of the hyperoctahedral group of order <math>n</math> a ''Type B descent'' of <math>\pi</math> is an index <math>i \in \{0,1,\ldots,n-1\}</math> for which <math>\pi(i) > \pi(i-1)</math>, with the convention that <math>\pi(0)=0</math>. The '''Type B Eulerian number''' <math>B(n,k)</math> is the number of elements of the hyperoctahedral group of order <math>n</math> with exactly <math>k</math> descents.<ref>{{cite journal |last1=Chow |first1=Chak-On |last2=Gessel |first2=Ira M. |title=On the descent numbers and major indices for the hyperoctahedral group |journal=Advances in Applied Mathematics |date=March 2007 |volume=38 |issue=3 |pages=275–301 |doi=10.1016/j.aam.2006.07.003}}</ref> They are given by the following formula:<ref>{{Cite OEIS|sequencenumber=A060187 |name=Triangle read by rows: Eulerian numbers of type B}}</ref> :<math>B(n,k)=\sum_{i=1}^k(-1)^{k-i}\binom{n}{k-i}(2i-1)^{n-1}.</math>

The table of <math>B(n,k)</math> (sequence [https://oeis.org/A060187 A060187] in the OEIS) is :{| class="wikitable" style="text-align:right;" |- ! {{diagonal split header|''n''&nbsp;|&nbsp;''k''}} ! width="50" | 0 ! width="50" | 1 ! width="50" | 2 ! width="50" | 3 ! width="50" | 4 ! width="50" | 5 |- ! 0 | 1 || || || || || |- ! 1 | 1 || 1 || || || || |- ! 2 | 1 || 6 || 1 || || || |- ! 3 | 1 || 23 || 23 || 1 || || |- ! 4 | 1 || 76 || 230 || 76 || 1 || |- ! 5 | 1 || 237 || 1682 || 1682 || 237 || 1 |}

The corresponding polynomials <math>M_n(x) = \sum_{k=0}^{n}B(n,k)x^k</math> are called '''midpoint Eulerian polynomials''' because of their use in interpolation and spline theory; see Schoenberg.<ref>{{cite journal |last1=Schoenberg |first1=I. J. |title=Cardinal Interpolation and Spline Functions IV. The Exponential Euler Splines |journal=Linear Operators and Approximation / Lineare Operatoren und Approximation |date=1972 |pages=382–404 |doi=10.1007/978-3-0348-7283-6_34 |isbn=978-3-0348-7285-0 }}</ref>

The Type B Eulerian numbers and polynomials satisfy many similar identities, and have many similar properties, as the Type A, i.e., usual, Eulerian numbers and polynomials. For example, for any <math>n \geq 1</math>, :<math>\sum_{i=0}^{\infty}(2i+1)^nx^i = \frac{M_{n}(x)}{(1-x)^{n+1}}.</math> And the Type B Eulerian numbers give the h-vector of the simple polytope dual to the Type B permutohedron.

In fact, one can define Eulerian numbers for any finite Coxeter group with analogous properties.{{sfnp|Petersen|2015|loc=Part III}}

==Eulerian numbers of the second order== The permutations of the multiset <math display="inline">\{1, 1, 2, 2, \ldots, n, n\}</math> 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 <math display="inline">(2n-1)!!</math>. These are called Stirling permutations.

The Eulerian number of the second order, denoted <math display="inline"> \left\langle \! \left\langle {n \atop m} \right\rangle \! \right\rangle </math>, counts the number of all such Stirling 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: :<math> \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, </math> with initial condition for ''n'' = 0, expressed in Iverson bracket notation: :<math> \left\langle \!\! \left\langle {0 \atop k} \right\rangle \!\! \right\rangle = [k=0].</math> Correspondingly, the Eulerian polynomial of second order, here denoted ''P''_''n'' (no standard notation exists for them) are :<math>P_n(x) := \sum_{k=0}^n \left\langle \!\! \left\langle {n \atop k} \right\rangle \!\! \right\rangle x^k </math> and the above recurrence relations are translated into a recurrence relation for the sequence ''P''_''n''(''x''): :<math> P_{n+1}(x) = (2nx+1) P_n(x) - x(x-1) P_n^\prime(x) </math> with initial condition <math> P_0(x) = 1 </math>. The latter recurrence may be written in a somewhat more compact form by means of an integrating factor: :<math> (x-1)^{-2n-2} P_{n+1}(x) = \left( x\,(1-x)^{-2n-1} P_n(x) \right)^\prime </math> so that the rational function :<math> u_n(x) := (x-1)^{-2n} P_{n}(x) </math> satisfies a simple autonomous recurrence: :<math> u_{n+1} = \left( \frac{x}{1-x} u_n \right)^\prime, \quad u_0=1</math> Whence one obtains the Eulerian polynomials of second order as <math display="inline">P_n(x) = (1-x)^{2n} u_n(x)</math>, and the Eulerian numbers of second order as their coefficients.

The Eulerian polynomials of the second order satisfy an identity analogous to the identity :<math>\sum_{i=1}^\infty i^n x^i = \frac{xA_n(x)}{(1-x)^{n+1}}</math> satisfied by the usual Eulerian polynomials. Specifically, as proved by Gessel and Stanley,<ref>{{cite journal |last1=Gessel |first1=Ira |last2=Stanley |first2=Richard P |title=Stirling polynomials |journal=Journal of Combinatorial Theory, Series A |date=1 January 1978 |volume=24 |issue=1 |pages=24–33 |doi=10.1016/0097-3165(78)90042-0}}</ref> they satisfy the identity :<math>\sum_{m=0}^{\infty}\left\{ {n+m \atop m} \right\}x^m = \frac{xP_n(x)}{(1-x)^{2n+1}}</math> where again the <math>\left\{ {n \atop k} \right\}</math> denote the Stirling numbers of the second kind. (This appearance of the Stirling numbers explains the terminology "Stirling permutations.")

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

:{| class="wikitable" style="text-align:right;" |- ! {{diagonal split header|''n''&nbsp;|&nbsp;''k''}} ! width="50" | 0 ! width="50" | 1 ! width="50" | 2 ! width="50" | 3 ! width="50" | 4 ! width="50" | 5 ! width="50" | 6 ! width="50" | 7 ! width="50" | 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 <math display="inline">P_n(1)</math>, is <math display="inline">(2n-1)!!</math>.

Indexing the second-order Eulerian numbers comes in three flavors: * {{OEIS|A008517}} following Riordan and Comtet, * {{OEIS|A201637}} following Graham, Knuth, and Patashnik, * {{OEIS|A340556}}, 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. * {{cite journal|first1=L. |last1=Carlitz|title=Eulerian Numbers and polynomials | year=1959|journal=Math. Mag.|volume=32 | number=5 | pages=247–260|doi=10.2307/3029225|jstor=3029225}} * {{cite book |last1=Comtet |first1=Louis |url=https://ia801306.us.archive.org/27/items/Comtet_Louis_-_Advanced_Coatorics/Comtet_Louis_-_Advanced_Combinatorics.pdf |title=Advanced Combinatorics: The Art of Finite and Infinite Expansions |year=1974 |publisher=Springer Netherlands |isbn=978-94-010-2198-2 |location=Dordrecht}} * {{cite journal|first1=H. W. |last1=Gould|title=Evaluation of sums of convolved powers using Stirling and Eulerian Numbers|year=1978|journal=Fib. Quart.|volume =16 |pages=488–497|number=6|doi=10.1080/00150517.1978.12430271 |url=https://www.fq.math.ca/16-6.html}} * {{cite journal|first1=Jacques | last1=Desarmenien | first2=Dominique | last2=Foata | title=The signed Eulerian numbers| journal=Discrete Math. |year=1992|volume=99 | number=1–3| pages=49–58|doi=10.1016/0012-365X(92)90364-L| doi-access=free}} * {{cite journal|year=1992|journal=Europ. J. Combinat. | volume=13 | pages=379–399|number=5|title= On the Eulerian Numbers M=max (A(n,k))|first1=Leonce |last1=Lesieur | first2=Jean-Louis | last2=Nicolas|doi=10.1016/S0195-6698(05)80018-6|doi-access=free}} * {{cite journal |first1=P. L. |last1=Butzer | first2=M. |last2=Hauss |title=Eulerian numbers with fractional order parameters |journal=Aequationes Mathematicae | year=1993 | volume=46|issue=1–2 |pages=119–142 |url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN002610515 | doi=10.1007/bf01834003 |s2cid=121868847 }} * {{cite journal|first1=M.V. |last1=Koutras|title=Eulerian numbers associated with sequences of polynomials|journal=Fib. Quart.|year=1994|pages=44–57|volume=32|number=1|doi=10.1080/00150517.1994.12429255 |url=https://www.fq.math.ca/32-1.html}} * {{cite book|last1=Graham|first1 = Ronald|author-link=Ronald Graham| last2=Knuth|first2=Donald|author-link2=Donald Knuth|last3=Patashnik|first3=Oren|author-link3=Oren Patashnik|year=1994|title=Concrete Mathematics: A Foundation for Computer Science|edition=2nd|publisher=Addison-Wesley|pages=267–272}} <!--See also http://www.combinatorics.org/Volume_4/PDF/v4i2r18.pdf --> *{{cite journal|first1=Leetsch C. |last1=Hsu |author1-link=Leetsch C. Hsu | first2=Peter | last2=Jau-Shyong Shiue| title=On certain summation problems and generalizations of Eulerian polynomials and numbers|journal=Discrete Math. | volume=204 | year=1999 |issue=1–3 | doi=10.1016/S0012-365X(98)00379-3|pages=237–247|doi-access=free }} * {{cite arXiv|eprint=0710.1124 |title=Apostol-Bernoulli functions, derivative polynomials and Eulerian polynomials|year=2007|first1=Khristo N. |last1=Boyadzhiev|class=math.CA}} * {{cite book |first1=T. Kyle|last1= Petersen|year=2015|title = Eulerian Numbers|series= Birkhäuser Advanced Texts Basler Lehrbücher|publisher=Birkhäuser|doi=10.1007/978-1-4939-3091-3_1|isbn= 978-1-4939-3090-6}}

== Citations == {{Reflist|colwidth=30em}}

==External links== *[http://oeis.org/wiki/Eulerian_polynomials Eulerian Polynomials] at OEIS Wiki. *{{MathPages|id=home/kmath012/kmath012|title=Eulerian Numbers}} *{{MathWorld|title=Eulerian Number|urlname=EulerianNumber}} *{{MathWorld|title=Euler's Number Triangle|urlname=EulersNumberTriangle}} *{{MathWorld|title=Worpitzky's Identity|urlname=WorpitzkysIdentity}} *{{MathWorld|title=Second-Order Eulerian Triangle|urlname=Second-OrderEulerianTriangle}} *[http://go.helms-net.de/math/binomial_new/01_12_Eulermatrix.pdf Euler-matrix] (generalized rowindexes, divergent summation)

Category:Enumerative combinatorics Category:Factorial and binomial topics Category:Integer sequences Category:Triangles of numbers