Search: id:A090582
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%I A090582
%S A090582 1,2,1,6,6,1,24,36,14,1,120,240,150,30,1,720,1800,1560,540,62,1,5040,
%T A090582 15120,16800,8400,1806,126,1,40320,141120,191520,126000,40824,5796,254,
%U A090582 1,362880,1451520,2328480,1905120,834120,186480,18150,510,1,3628800
%N A090582 Numerator Q(m,n) of probability P(m,n)=Q(m,n)/n^m to see each card at 
               least once if m>=n cards are drawn with replacement from a deck of 
               n cards, written in a two-dimensional array read by antidiagonals 
               with Q(m,m) as first row and Q(m,1) as first column.
%C A090582 The sequence is given as a matrix with the first row containing the cases 
               #draws=size_of_deck. The second row contains #draws=1+size_of_deck. 
               If "mn" indicates m cards drawn from a deck with n cards then the 
               locations in the matrix are:
%C A090582 11 22 33 44 55 66 77 ...
%C A090582 21 32 43 54 65 76 87 ...
%C A090582 31 42 53 64 75 86 97 ...
%C A090582 41 52 63 74 85 .. .. ...
%C A090582 read by antidiagonals ->:
%C A090582 11, 22, 21, 33, 32, 31, 44, 43, 42, 41, 55, 54, 53, 52, ....
%C A090582 The probabilities are given by Q(m,n)/n^m:
%C A090582 .(m,n):.....11..22..21..33..32..31..44..43..42..41...55...54..53..52..51
%C A090582 .....Q:......1...2...1...6...6...1..24..36..14...1..120..240.150..30...1
%C A090582 ...n^m:......1...4...1..27...8...1.256..81..16...1.3125.1024.243..32...1
%C A090582 %.Success:.100..50.100..22..75.100...9..44..88.100....4...23..62..94.100
%C A090582 P(n,n) = n!/n^n which can be approximated by sqrt(pi*(2n+1/3))/e^n (Gosper's 
               approximation to n!).
%C A090582 Triangle T(n,k), 1<=k<=n, read by rows given by [1, 1, 2, 2, 3, 3, 4, 
               4, 5, 5, ...] DELTA [[0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 6, 0, ...] 
               where DELTA is the operator defined in A084938 . - Philippe DELEHAM 
               (kolotoko(AT)wanadoo.fr), Nov 10 2006
%D A090582 See A073593.
%H A090582 Kevin Buhr, Michael Press and DMB, <a href="http://mathforum.org/discuss/
               sci.math/t/435082">Collecting a deck of cards on the street.</a> 
               Thread in NG sci.math.
%H A090582 Hugo Pfoertner, <a href="http://www.randomwalk.de/sequences/a090582.txt">
               "Hit all" probabilities: Program and results.</a>
%F A090582 Q(m, n) = sum_(k=0..n-1) (-1)^k * C(n, k) * (n-k)^m where C(n, k) is 
               the binomial coefficient "n choose k".
%F A090582 Generated by Stirling numbers of the second kind multiplied by a factorial. 
               - Victor Adamchik, Oct 05 2005
%F A090582 Formulae from Tom Copeland (tcjpn(AT)msn.com), Oct 07 2008: (Start)
%F A090582 G(x,t) = 1/ {1 + [1-exp(x t)]/t} = 1 + 1 x + (2 + t) x^2/2! + (6 + 6t 
               + t^2) x^3/3! + ...
%F A090582 gives row polynomials of A090582-- reverse f-polynomials for the permutohedra 
               (see A019538).
%F A090582 G(x,t-1) = 1 + 1 x + (1 + t) x^2 / 2! + (1 + 4t + t^2) x^3 / 3! + ...
%F A090582 gives row polynomials for A008292, the h-polynomials for permutohedra.
%F A090582 G[(t+1)x,-1/(t+1)] = 1 + (1+ t) x + (1 + 3t + 2 t^2) x^2 / 2! + ...
%F A090582 gives row polynomials of A028246. (End)
%t A090582 In[1]:= Table[Table[k! StirlingS2[n, k], {k, n, 1, -1}], {n, 1, 6}] - 
               Victor Adamchik, Oct 05 2005
%o A090582 FORTRAN program given at Pfoertner link.
%Y A090582 Cf. A073593 first m>=n giving at least 50% probability, A085813 ditto 
               for 95%, A055775 n^n/n!, A090583 Gosper's approximation to n!.
%Y A090582 Reflected version of A019538.
%Y A090582 Sequence in context: A105278 A008297 A048999 this_sequence A079641 A075181 
               A052121
%Y A090582 Adjacent sequences: A090579 A090580 A090581 this_sequence A090583 A090584 
               A090585
%K A090582 frac,nonn,tabl
%O A090582 1,2
%A A090582 Hugo Pfoertner (hugo(AT)pfoertner.org), Jan 11 2004

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