emmy.special.factorial — org.mentat/emmy 0.31.0

Namespace holding implementations of variations on the factorial function.

->bigint cljs

(->bigint x)

If x is a fixed-precision integer, returns a emmy.util/bigint version of x. Else, acts as identity.

This is useful in cases where you may want to multiply x by other large numbers, but don't want to try and convert something that can't overflow, like a symbol, into bigint.

bell clj/s

(bell n)

Returns the nth Bell number, i.e., the number of ways a set of n elements can be partitioned into nonempty subsets.

The nth Bell number is denoted $B_n$.

binomial-coefficient clj/s

(binomial-coefficient n k)

Returns the binomial coefficient, i.e., the coefficient of the $x^k$ term in the polynomial expansion of the binomial power $(1 + x)^n$.

This quantity is sometimes pronounced "n choose k".

For negative n or k, binomial-coefficient matches the behavior provided by Mathematica, described at this page. Given negative n, returns

;; for k >= 0
(* (expt -1 k)
   (binomial-coefficient (+ (- n) k -1) k))

;; for k >= 0
(* (expt -1 (- n k))
   (binomial-coefficient (+ (- k) -1) (- n k)))

;; otherwise:
0

double-factorial clj/s

(double-factorial n)

Returns the product of all integers from 1 up to n that have the same parity (odd or even) as n.

([[double-factorial]] 0) is defined as an empty product and equal to 1.

double-factorial with argument n is equivalent to ([[multi-factorial]] n 2), but slightly more general in that it can handle negative values of n.

If n is negative and even, returns ##Inf.

If n is negative and odd, returns (/ (double-factorial (+ n 2)) (+ n 2)).

For justification, see the Wikipedia page on the extension of double factorial to negative arguments.

factorial clj/s

(factorial n)

Returns the factorial of n, i.e., the product of 1 to n (inclusive).

factorial will return a platform-specific emmy.util/bigint given some n that causes integer overflow.

factorial-power clj/s multimethod

Alias for falling-factorial.

falling-factorial clj/s multimethod

(falling-factorial a b)

generic falling-factorial.

Returns the falling factorial, of a to the b, defined as the polynomial

$$(a)_b = a^{\underline{b}} = a(a - 1)(a - 2) \cdots (a - b - 1)$$

Given a negative b, ([[falling-factorial]] a b) is equivalent to (invert ([[rising-factorial]] (inc a) (- b))), or ##Inf if the denominator evaluates to 0.

The coefficients that appear in the expansions of falling-factorial called with a symbolic first argument and positive integral second argument are the Stirling numbers of the first kind (see stirling-first-kind).

multi-factorial clj/s

(multi-factorial n k)

Returns the product of the positive integers up to n that are congruent to (mod n k).

When k equals 1, equivalent to ([[factorial]] n).

See the [Wikipedia page on generalizations of double-factorial](https://en.wikipedia.org/wiki/Double_factorial#Generalizations) for more detail.

If you need to extend multi-factorial to negative n or k, that page has suggestions for generalization.

pochhammer clj/s multimethod

Alias for falling-factorial.

rising-factorial clj/s multimethod

(rising-factorial a b)

generic rising-factorial.

Returns the rising factorial, of a to the b, defined as the polynomial

$$(a)^b = a^{\overline{b}} = a(a + 1)(a + 2) \cdots (a + b - 1)$$

Given a negative b, ([[rising-factorial]] a b) is equivalent to (invert ([[falling-factorial]] (dec a) (- b))), or ##Inf if the denominator evaluates to 0.

stirling-first-kind clj/s

(stirling-first-kind n k & {:keys [unsigned?]})

Given n and k, returns the number of permutations of n elements which contain exactly k permutation cycles. This is called the Stirling number s(n, k) of the first kind.

By default, returns the signed Stirling number of the first kind. Pass the :unsigned? true keyword option to retrieve the signed Stirling number. (Or take the absolute value of the result...)

(stirling-first-kind 13 2)
;;=> -1486442880

(stirling-first-kind 13 2 :unsigned? true)
;;=> 1486442880

stirling-second-kind clj/s

(stirling-second-kind n k)

Returns $S(n,k)$, the number of ways to partition a set of n objects into k non-empty subsets.

This is called a Stirling number of the second kind.

subfactorial clj/s

(subfactorial n)

Returns the number of permutations of n objects in which no object appears in its original position. (Each of these permutations is called a 'derangement' of the set.)

References

• Subfactorial page at Wolfram Mathworld
• John Cook, Variations on Factorial
• John Cook, Subfactorial
• 'Derangement' on Wikipedia