

A217111


Number of pandigital numbers <= 10^n.


1



0, 0, 0, 0, 0, 0, 0, 0, 0, 3265920, 182891520, 5751285120, 134183589120, 2592611400960, 43947813288960, 676736110229760, 9685234777397760, 130890592784891520, 1689704521363998720, 21016063609130056320, 253507542701850904320, 2981020379966298432000
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OFFSET

1,10


COMMENTS

The number of numbers with <= n places which contain all decimal digits 0..9.
The ratio a(n)/10^n indicates the relative proportion of pandigital numbers <= 10^n compared to all numbers <= 10^n. Since that ratio converges to the limit 1 for n > infinity this can be expressed for large numbers as follows (in a slightly popular manner): ‘Almost all numbers contain all decimal digits 0..9’.
Example: a(n)/10^n = 0. 99973107526479... for n = 100; in this case 99.9731…% of all numbers <= 10^100 contain all digits 0..9. Conversely, only the tiny proportion of 0.000268924735210... (<0.03%) lacks at least one digit. That’s astonishing! Intuitively, this is not what one would expect. In fact, for smaller numbers (with which most people are faced normally) the relative portion of numbers which missing at least one digit is significantly larger. Of course, for n < 10 the portion is 100%, and even for numbers <= 10^10 or <= 10^20 the relative proportion of numbers which contain not all digits 0..9 is 99.96734...% or 78.98393...%, respectively. 10^27 is the least power of 10 such that the pandigital numbers hold the majority. Here, the proportion of pandigital numbers amongst all numbers <= 10^27 is 51.50961...%. So one could bet that a randomly chosen number <= 10^27 contains all digits.
Partial sums of A217110.


LINKS

Hieronymus Fischer, Table of n, a(n) for n = 1..200


FORMULA

a(n) = 9*9!*sum_{j=1..n} S2(j,10), where the S2(j,10) are the Stirling numbers of the second kind (cf. triangle A008277).
Asymptotic behavior:
lim a(n)/10^n = 1, for n > infinity.
G.f.: g(x) = 9*9!*x^10/((1x)*product_{j=1..10} (1jx)).


EXAMPLE

a(k) = 0, for k < 10 since there a no pandigital numbers <= 10^9, trivially.
a(10) = 9*9!, since the first digit can be in the range 1..9 and for the following 9 digits there are 9, 8, 7, ..., 1 possible values.


CROSSREFS

Cf. A171102, A050278, A011540, A002542, A053283, A217094, A217110.
Sequence in context: A199634 A203987 A217110 * A036471 A206316 A186959
Adjacent sequences: A217108 A217109 A217110 * A217112 A217113 A217114


KEYWORD

nonn,base


AUTHOR

Hieronymus Fischer, Feb 13 2013


STATUS

approved



