Pequeño pingüino

See also: Beautiful

The numbers of Beautiful , which bear the name of Eric Temple Beautiful, often meet in Combinatoire. These numbers form a succession of entireties which starts as follows:

B_0=1, \ quad B_1=1, \ quad B_2=2, \ quad B_3=5, \ quad B_4=15, \ quad B_5=52, \ quad B_6=203, \ quad \ dots

(A000110 continuation in the electronic Encyclopedia of the whole continuations)

In general, B N is the number of partitions of a whole of cardinal N . ( B 0 is equal to 1 because there is exactly a partition of the empty set. A partition of a unit E is by definition a whole of nonempty and disjoined parts two to two, whose meeting is equal to the unit E .) Each part of a partition of the empty set is part of the empty set thus is empty (that is obvious), and their meeting is equal to the empty set. Therefore, the singleton empty set is the only partition of the empty set.

The numbers of Beautiful satisfy the formula of recurrence:

B_ {n+1} = \ sum_ {k=0} ^ {N} {C_n^k B_k}.
(where C_n^k is a binomial Coefficient)

They also satisfy the formula of Dobinski :

B_n= \ frac {1} {E} \ sum_ {k=0} ^ \ infty \ frac {k^n} {K!}
who is the moment of order N of a law of Poisson of parameter 1.

They also satisfy the congruence of Touchard : if p is a Prime number unspecified then

B_ {p+n} \ equiv B_n+B_ {n+1} \ (p).
(relation of congruence modulo p )

Each number of Beautiful is a sum of the '' numbers of Stirling of second species ''

B_n= \ sum_ {k=1} ^n S (N, K).

The generating series exponential of the numbers of Beautiful is

e^ {(e^x-1)}= \ sum_ {n=0} ^ \ infty \ frac {B_nx^n} {N!}=1+x+2 \ frac {x^2} {2!}+5 \ frac {x^3} {3!} + 15 \ frac {x^4} {4!} + \ dots

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