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A059060 Card-matching numbers (Dinner-Diner matching numbers). 4
1, 0, 0, 0, 0, 1, 1, 0, 16, 0, 36, 0, 16, 0, 1, 346, 1824, 4536, 7136, 7947, 6336, 3936, 1728, 684, 128, 48, 0, 1, 748521, 3662976, 8607744, 12880512, 13731616, 11042688, 6928704, 3458432, 1395126, 453888, 122016, 25344, 4824, 512, 96, 0, 1, 3993445276 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,9

COMMENTS

This is a triangle of card matching numbers. A deck has n kinds of cards, 4 of each kind. The deck is shuffled and dealt in to n hands with 4 cards each. 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..4n). The probability of exactly k matches is T(n,k)/((4n)!/(4!)^n).

Rows have lengths 1,5,9,13,...

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.

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, k is the number of cards of each kind (here k is 4) 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 j-th coefficient on x of the rook polynomial.

EXAMPLE

There are 16 ways of matching exactly 2 cards when there are 2 different kinds of cards, 4 of each so T(2,2)=16.

From Joerg Arndt, Nov 08 2020: (Start)

The first few rows are

1

0, 0, 0, 0, 1

1, 0, 16, 0, 36, 0, 16, 0, 1

346, 1824, 4536, 7136, 7947, 6336, 3936, 1728, 684, 128, 48, 0, 1

748521, 3662976, 8607744, 12880512, 13731616, 11042688, 6928704, 3458432, 1395126, 453888, 122016, 25344, 4824, 512, 96, 0, 1

3993445276, 18743463360, 42506546320, 61907282240, 64917874125, 52087325696, 33176621920, 17181584640, 7352761180, 2628808000, 790912656, 201062080, 43284010, 7873920, 1216000, 154496, 17640, 1280, 160, 0, 1 (End)

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 5 do seq(coeff(f(t, n, 4), t, m)/4!^n, m=0..4*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, n, 4], t, m]/4!^n, {n, 0, 5}, {m, 0, 4*n}] // Flatten (* Jean-Fran├žois Alcover, Feb 22 2013, translated from Maple *)

CROSSREFS

Cf. A008290, A059056-A059071.

Sequence in context: A326852 A070570 A005077 * A331140 A059681 A135925

Adjacent sequences:  A059057 A059058 A059059 * A059061 A059062 A059063

KEYWORD

nonn,tabf,nice

AUTHOR

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

STATUS

approved

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Last modified February 27 10:25 EST 2021. Contains 341649 sequences. (Running on oeis4.)