Japan stirlingses Reservation Number %F0%9F%93%9E850.308.3021%F0%9F%93%9E
(0.007 seconds)
11—20 of 512 matching pages
11: 26.13 Permutations: Cycle Notation
…
►The Stirling cycle numbers of the first kind, denoted by , count the number of permutations of with exactly cycles.
They are related to Stirling numbers of the first kind by
…See §26.8 for generating functions, recurrence relations, identities, and asymptotic approximations.
…
►The derangement number, , is the number of elements of with no fixed points:
…
►A permutation is even or odd according to the parity of the number of transpositions.
…
12: Guide to Searching the DLMF
…
►
term:
►
phrase:
…
►
proximity operator:
…
►
►
…
a textual word, a number, or a math symbol.
any double-quoted sequence of textual words and numbers.
adj, prec/n, and near/n, where n is any positive natural number.
$ |
stands for any number of alphanumeric characters |
---|---|
(the more conventional * is reserved for the multiplication operator) |
|
… |
13: 26.6 Other Lattice Path Numbers
§26.6 Other Lattice Path Numbers
… ►Delannoy Number
… ►Motzkin Number
… ►Narayana Number
… ►§26.6(iv) Identities
…14: 16.7 Relations to Other Functions
…
►For , , symbols see Chapter 34.
Further representations of special functions in terms of functions are given in Luke (1969a, §§6.2–6.3), and an extensive list of functions with rational numbers as parameters is given in Krupnikov and Kölbig (1997).
15: 26.11 Integer Partitions: Compositions
…
►
denotes the number of compositions of , and is the number of compositions into exactly
parts.
is the number of compositions of with no 1’s, where again .
…
►The Fibonacci numbers are determined recursively by
…
►
26.11.6
.
…
►Additional information on Fibonacci numbers can be found in Rosen et al. (2000, pp. 140–145).
16: 26.5 Lattice Paths: Catalan Numbers
§26.5 Lattice Paths: Catalan Numbers
►§26.5(i) Definitions
► is the Catalan number. … ►§26.5(ii) Generating Function
… ►§26.5(iii) Recurrence Relations
…17: 27.2 Functions
…
►
§27.2(i) Definitions
… ►where are the distinct prime factors of , each exponent is positive, and is the number of distinct primes dividing . … ►(See Gauss (1863, Band II, pp. 437–477) and Legendre (1808, p. 394).) … ► … ►§27.2(ii) Tables
…18: 26.14 Permutations: Order Notation
…
►As an example, is an element of The inversion number is the number of pairs of elements for which the larger element precedes the smaller:
…
►
►The Eulerian number, denoted , is the number of permutations in with exactly descents.
…The Eulerian number
is equal to the number of permutations in with exactly excedances.
…
►
§26.14(iii) Identities
…19: 26.7 Set Partitions: Bell Numbers
§26.7 Set Partitions: Bell Numbers
►§26.7(i) Definitions
… ►§26.7(ii) Generating Function
… ►§26.7(iii) Recurrence Relation
… ►§26.7(iv) Asymptotic Approximation
…20: 34.10 Zeros
…
►In a symbol, if the three angular momenta do not satisfy the triangle conditions (34.2.1), or if the projective quantum numbers do not satisfy (34.2.3), then the symbol is zero.
…However, the and symbols may vanish for certain combinations of the angular momenta and projective quantum numbers even when the triangle conditions are fulfilled.
Such zeros are called nontrivial zeros.
►For further information, including examples of nontrivial zeros and extensions to symbols, see Srinivasa Rao and Rajeswari (1993, pp. 133–215, 294–295, 299–310).