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A059066 Card-matching numbers (Dinner-Diner matching numbers). 0
1, 2, 3, 0, 1, 10, 24, 27, 16, 12, 0, 1, 56, 216, 378, 435, 324, 189, 54, 27, 0, 1, 346, 1824, 4536, 7136, 7947, 6336, 3936, 1728, 684, 128, 48, 0, 1, 2252, 15150, 48600, 99350, 144150, 156753, 131000, 87075, 45000, 19300, 6000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

This is a triangle of card matching numbers. A deck has 3 kinds of cards, n of each kind. The deck is shuffled and dealt in to 3 hands each with n cards. A match occurs for every card in the j-th hand of kind j. Triangle T(n,k) is the number of ways of achieving exactly k matches (k=0..3n). The probability of exactly k matches is T(n,k)/((3n)!/n!^3).

Rows have lengths 1,4,7,10,...

Analogous to A008290. - Zerinvary Lajos, Jun 22 2005

REFERENCES

F. N. David and D. E. Barton, Combinatorial Chance, Hafner, NY, 1962, Ch. 7 and Ch. 12.

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, pp. 174-178.

R. P. Stanley, Enumerative Combinatorics Volume I, Cambridge University Press, 1997, p. 71.

LINKS

Table of n, a(n) for n=0..45.

D. Callan, A combinatorial interpretation for an identity of Barrucand, JIS 11 (2008) 08.3.4.

F. F. Knudsen and I. Skau, On the Asymptotic Solution of a Card-Matching Problem, Mathematics Magazine 69 (1996), 190-197.

Barbara H. Margolius, Dinner-Diner Matching Probabilities

B. H. Margolius, The Dinner-Diner Matching Problem, Mathematics Magazine, 76 (2003), 107-118.

S. G. Penrice, Derangements, permanents and Christmas presents, The American Mathematical Monthly 98(1991), 617-620.

Index entries for sequences related to card matching

FORMULA

G.f.: sum(coeff(R(x, n, k), x, j)*(t-1)^j*(n*k-j)!, j=0..n*k) where n is the number of kinds of cards (3 in this case), k is the number of cards of each kind and R(x, n, k) is the rook polynomial given by R(x, n, k)=(k!^2*sum(x^j/((k-j)!^2*j!))^n (see Stanley or Riordan). coeff(R(x, n, k), x, j) indicates the coefficient for x^j of the rook polynomial.

EXAMPLE

There are 27 ways of matching exactly 2 cards when there are 2 cards of each kind and 3 kinds of card so T(2,2)=27.

MAPLE

p := (x, k)->k!^2*sum(x^j/((k-j)!^2*j!), j=0..k); R := (x, n, k)->p(x, k)^n; f := (t, n, k)->sum(coeff(R(x, n, k), x, j)*(t-1)^j*(n*k-j)!, j=0..n*k);

for n from 0 to 7 do seq(coeff(f(t, 3, n), t, m)/n!^3, m=0..3*n); od;

MATHEMATICA

p[x_, k_] := k!^2*Sum[ x^j/((k-j)!^2*j!), {j, 0, k}]; r[x_, n_, k_] := p[x, k]^n; f[t_, n_, k_] := Sum[ Coefficient[r[x, n, k], x, j]*(t-1)^j*(n*k-j)!, {j, 0, n*k}]; Table[ Coefficient[ f[t, 3, n], t, m]/n!^3, {n, 0, 5}, {m, 0, 3*n}] // Flatten (* Jean-Fran├žois Alcover, Mar 04 2013, translated from Maple *)

CROSSREFS

Cf. A008290, A059056-A059071.

Cf. A008290.

Sequence in context: A323883 A008290 A322147 * A059067 A065861 A126832

Adjacent sequences:  A059063 A059064 A059065 * A059067 A059068 A059069

KEYWORD

nonn,tabf,nice

AUTHOR

Barbara Haas Margolius (margolius(AT)math.csuohio.edu)

STATUS

approved

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Last modified December 11 13:28 EST 2019. Contains 329916 sequences. (Running on oeis4.)