“Combination Lock”的概念、定义、习题解题方法及翻译-数学辞典

单词

Combination Lock

释义

Combination Lock

Let a combination of buttons be a Sequence of disjoint nonempty Subsets of the Set . If the number of possible combinations is denoted , then satisfies the RecurrenceRelation

(1)

with

. This can also be written

(2)

where the definition

has been used. Furthermore,

(3)

where

are Eulerian Numbers. In terms of the Stirling Numbers of the SecondKind

(4)

can also be given in closed form as

(5)

where

is the Polylogarithm. The first few values of

for

, 2, ... are 1, 3, 13, 75, 541,4683, 47293, 545835, 7087261, 102247563, ... (Sloane's A000670).

The quantity

(6)

satisfies the inequality

(7)

References

Sloane, N. J. A. SequenceA000670/M2952in ``An On-Line Version of the Encyclopedia of Integer Sequences.''http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S.The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.

Velleman, D. J. and Call, G. S. ``Permutations and Combination Locks.'' Math. Mag. 68, 243-253, 1995.

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