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21: 26.7 Set Partitions: Bell Numbers

§26.7 Set Partitions: Bell Numbers

… ► $B\left(n\right)$ is the number of partitions of $\{1,2,\mathrm{\dots },n\}$. …

22: 27.13 Functions

… ►

§27.13(i) Introduction

… ►Each representation of $n$ as a sum of elements of $S$ is called a partition of $n$, and the number $S(n)$ of such partitions is often of great interest. …

23: 3.7 Ordinary Differential Equations

… ►Assume that we wish to integrate (3.7.1) along a finite path $𝒫$ from $z=a$ to $z=b$ in a domain $D$. The path is partitioned at $P+1$ points labeled successively $z_0,z_1,\mathrm{\dots },z_P$, with $z_0=a$, $z_P=b$. … ►

3.7.10

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Defines:

$𝐀(\tau ,z)$: matrix (locally)

Symbols:

${\tau }_j$: change of variable and $P$: partitioning point

Permalink:

http://dlmf.nist.gov/3.7.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §3.7(iii), §3.7 and Ch.3

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3.7.11 $$𝐰={[w(z_0),w^{\prime }(z_0),w(z_1),w^{\prime }(z_1),\mathrm{\dots },w(z_P),w^{\prime }(z_P)]}^{\mathrm{T}},$$

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Symbols:

$w(z)$: function and $P$: partitioning point

Permalink:

http://dlmf.nist.gov/3.7.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §3.7(iii), §3.7 and Ch.3

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3.7.13 $${𝐀}_P𝐰=𝐛.$$

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Symbols:

$𝐀(\tau ,z)$: matrix, $𝐛(\tau ,z)$: vector and $P$: partitioning point

Referenced by:

§3.7(iii), §3.7(iv)

Permalink:

http://dlmf.nist.gov/3.7.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §3.7(iii), §3.7 and Ch.3

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24: 26.8 Set Partitions: Stirling Numbers

§26.8 Set Partitions: Stirling Numbers

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26.8.3 $n>k\ge 1$.

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Symbols:

$s(n,k)$: Stirling number of the first kind, $k$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/26.8.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §26.8(i), §26.8 and Ch.26

► $S(n,k)$ denotes the Stirling number of the second kind: the number of partitions of $\{1,2,\mathrm{\dots },n\}$ into exactly $k$ nonempty subsets. … ►

25: Bibliography R

… ►

On the partition function p(n). Proc. London Math. Soc. (2) 43 (4), pp. 241–254.

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External Links:

Document, MathReview Entry, ZentralBlatt

Referenced by:

§27.14(iii).

Encodings:

BibTeX

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Congruence properties of partitions. Math. Z. 9 (1-2), pp. 147–153.

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External Links:

ISSN 0025-5874, ISSN 0025-5874, Document, MathReview Entry, ZentralBlatt

Referenced by:

§27.14(v).

Encodings:

BibTeX

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Research Institute for Symbolic Computation, Hagenberg im Mühlkreis, Austria.

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Notes:

Software packages for hypergeometric and $q$-hypergeometric summation, sequences and power series, permutation groups, partition analysis, and difference equations.

External Links:

RISC Combinatorics Group

Referenced by:

6th item.

Encodings:

BibTeX

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26: Bibliography

… ►

On basic hypergeometric series, mock theta functions, and partitions. II. Quart. J. Math. Oxford Ser. (2) 17, pp. 132–143.

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External Links:

Document, MathReview (B. R. Bhonsle), ZentralBlatt

Referenced by:

§17.6(ii).

Encodings:

BibTeX

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Shanks’ convergence acceleration transform, Padé approximants and partitions. J. Combin. Theory Ser. A 43 (1), pp. 70–84.

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External Links:

ISSN 0097-3165, Document, MathReview (Kenneth A. Jukes), ZentralBlatt

Referenced by:

§17.18.

Encodings:

BibTeX

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The Theory of Partitions. Encyclopedia of Mathematics and its Applications, Vol. 2, Addison-Wesley Publishing Co., Reading, MA-London-Amsterdam.

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Notes:

Reprinted by Cambridge University Press, Cambridge, 1998.

External Links:

MathReview (L. Carlitz), ZentralBlatt

Referenced by:

§17.18, §17.2(iii), §17.2(vi), §17.5, Ch.17, §26.10(ii), §26.10(iii), §26.10(iv), §26.10(v), §26.10(vi), §26.11, §26.12(iv), §26.14(i), §26.14(ii), §26.16, §26.16, §26.21, §26.9(i), §26.9(i), §26.9(ii), §26.9(iv), §27.14(vi), Ch.27.

Encodings:

BibTeX

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Plane partitions. III. The weak Macdonald conjecture. Invent. Math. 53 (3), pp. 193–225.

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External Links:

ISSN 0020-9910, Document, MathReview (Edward A. Bender), ZentralBlatt

Referenced by:

§26.12(i), §26.12(ii).

Encodings:

BibTeX

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Continuous $q$-Hermite Polynomials when $q>1$ . In $q$-series and Partitions (Minneapolis, MN, 1988), IMA Vol. Math. Appl., Vol. 18, pp. 151–158.

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External Links:

MathReview (Walter Van Assche), ZentralBlatt

Referenced by:

§18.28(vii).

Encodings:

BibTeX

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27: Bibliography G

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Tables of Partitions. Royal Society Math. Tables, Vol. 4, Cambridge University Press.

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External Links:

ZentralBlatt

Referenced by:

§27.21.

Encodings:

BibTeX

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A table of partitions. Proc. London Math. Soc. (2) 39, pp. 142–149.

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External Links:

Document, ZentralBlatt

Referenced by:

§27.21.

Encodings:

BibTeX

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A table of partitions (II). Proc. London Math. Soc. (2) 42, pp. 546–549.

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External Links:

Document, ZentralBlatt

Referenced by:

§27.21.

Encodings:

BibTeX

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28: 35.1 Special Notation

… ► ►►►

$a,b$	complex variables.
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${\left[a\right]}_{\kappa }$	partitional shifted factorial (§35.4(i)).
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…

29: Bibliography L

… ►

A partition function connected with the modulus five. Duke Math. J. 8 (4), pp. 631–655.

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External Links:

Document, MathReview (R. D. James), ZentralBlatt

Referenced by:

§17.18.

Encodings:

BibTeX

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Lie algebraic approaches to classical partition identities. Adv. in Math. 29 (1), pp. 15–59.

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External Links:

ISSN 0001-8708, Document, MathReview (S. I. Gel’fand), ZentralBlatt

Referenced by:

§17.16.

Encodings:

BibTeX

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30: Bibliography O

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Distribution of the partition function modulo $m$ . Ann. of Math. (2) 151 (1), pp. 293–307.

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External Links:

ISSN 0003-486X, FullText, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

§27.14(v).

Encodings:

BibTeX

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