1,2

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:

11 22 33 44 55 66 77 ...

21 32 43 54 65 76 87 ...

31 42 53 64 75 86 97 ...

41 52 63 74 85 .. .. ...

read by antidiagonals ->:

11, 22, 21, 33, 32, 31, 44, 43, 42, 41, 55, 54, 53, 52, ....

The probabilities are given by Q(m,n)/n^m:

.(m,n):.....11..22..21..33..32..31..44..43..42..41...55...54..53..52..51

.....Q:......1...2...1...6...6...1..24..36..14...1..120..240.150..30...1

...n^m:......1...4...1..27...8...1.256..81..16...1.3125.1024.243..32...1

%.Success:.100..50.100..22..75.100...9..44..88.100....4...23..62..94.100

P(n,n) = n!/n^n which can be approximated by sqrt(pi*(2n+1/3))/e^n (Gosper's approximation to n!).

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

See A073593.

Kevin Buhr, Michael Press and DMB, Collecting a deck of cards on the street. Thread in NG sci.math.

Hugo Pfoertner, "Hit all" probabilities: Program and results.

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".

Generated by Stirling numbers of the second kind multiplied by a factorial. - Victor Adamchik, Oct 05 2005

Formulae from Tom Copeland (tcjpn(AT)msn.com), Oct 07 2008: (Start)

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! + ...

gives row polynomials of A090582-- reverse f-polynomials for the permutohedra (see A019538).

G(x,t-1) = 1 + 1 x + (1 + t) x^2 / 2! + (1 + 4t + t^2) x^3 / 3! + ...

gives row polynomials for A008292, the h-polynomials for permutohedra.

G[(t+1)x,-1/(t+1)] = 1 + (1+ t) x + (1 + 3t + 2 t^2) x^2 / 2! + ...

gives row polynomials of A028246. (End)

In[1]:= Table[Table[k! StirlingS2[n, k], {k, n, 1, -1}], {n, 1, 6}] - Victor Adamchik, Oct 05 2005

FORTRAN program given at Pfoertner link.

Cf. A073593 first m>=n giving at least 50% probability, A085813 ditto for 95%, A055775 n^n/n!, A090583 Gosper's approximation to n!.

Reflected version of A019538.

Sequence in context: A105278 A008297 A048999 this_sequence A079641 A075181 A052121

Adjacent sequences: A090579 A090580 A090581 this_sequence A090583 A090584 A090585

frac,nonn,tabl

Hugo Pfoertner (hugo(AT)pfoertner.org), Jan 11 2004