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A059071 Card-matching numbers (Dinner-Diner matching numbers) for 5 kinds of cards. 22
1, 44, 45, 20, 10, 0, 1, 440192, 975360, 1035680, 696320, 329600, 114176, 31040, 5120, 1280, 0, 32, 52097831424, 179811290880, 298276007040, 315423836640, 237742646400, 135296008128, 60059024640 (list; graph; refs; listen; history; text; internal format)



This is a triangle of card matching numbers. Two decks each have 5 kinds of cards, n of each kind. The first deck is laid out in order. The second deck is shuffled and laid out next to the first. A match occurs if a card from the second deck is next to a card of the same kind from the first deck. Triangle T(n,k) is the number of ways of achieving exactly k matches (k=0..5n). The probability of exactly k matches is T(n,k)/(5n)!.

Rows are of length 1,6,11,16,... = 5n+1 = A016861(n). - M. F. Hasler, Sep 20 2015


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.


Table of n, a(n) for n=0..24.

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

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

Barbara H. Margolius, Dinner-Diner Matching Probabilities

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

Index entries for sequences related to card matching


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 (5 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.


There are 1,035,680 ways of matching exactly 2 cards when there are 2 cards of each kind and 5 kinds of card so T(2,2)=1,035,680.


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 3 do seq(coeff(f(t, 5, n), t, m), m=0..5*n); od;


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, 5, n], t, m], {n, 0, 3}, {m, 0, 5*n}] // Flatten (* Jean-Fran├žois Alcover, Mar 04 2013, translated from Maple *)


Cf. A008290, A059056-A059071.

Sequence in context: A239534 A171906 A059070 * A085519 A112815 A242934

Adjacent sequences:  A059068 A059069 A059070 * A059072 A059073 A059074




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


Edited by M. F. Hasler, Sep 20 2015



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Last modified July 31 21:57 EDT 2021. Contains 346377 sequences. (Running on oeis4.)