Expected Number of Cycles of Any Length of A Random Permutation
We construct the bivariate generating function using, where is one for all cycles (every cycle contributes one to the total number of cycles).
Note that has the closed form
and generates the unsigned Stirling numbers of the first kind.
We have
Hence the expected number of cycles is, or about .
Read more about this topic: Random Permutation Statistics
Famous quotes containing the words expected, number, cycles, length and/or random:
“Between us two its not a star at all.
Its a new patented electric light,
Put up on trial by that Jerseyite
So much is being now expected of....”
—Robert Frost (18741963)
“Black lady,
what will I do
without your two flowers?
I have inhabited you, number by number.
I have pushed you in and out like a needle.”
—Anne Sexton (19281974)
“The stars which shone over Babylon and the stable in Bethlehem still shine as brightly over the Empire State Building and your front yard today. They perform their cycles with the same mathematical precision, and they will continue to affect each thing on earth, including man, as long as the earth exists.”
—Linda Goodman (b. 1929)
“With the ancient is wisdom; and in length of days understanding.”
—Bible: Hebrew Job, 12:12.
“Man always made, and still makes, grotesque blunders in selecting and measuring forces, taken at random from the heap, but he never made a mistake in the value he set on the whole, which he symbolized as unity and worshipped as God. To this day, his attitude towards it has never changed, though science can no longer give to force a name.”
—Henry Brooks Adams (18381918)