

A059071


Cardmatching numbers (DinnerDiner 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
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OFFSET

0,2


COMMENTS

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


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. 174178.
R. P. Stanley, Enumerative Combinatorics Volume I, Cambridge University Press, 1997, p. 71.


LINKS

Table of n, a(n) for n=0..24.
F. F. Knudsen and I. Skau, On the Asymptotic Solution of a CardMatching Problem, Mathematics Magazine 69 (1996), 190197.
B. H. Margolius, The DinnerDiner Matching Problem, Mathematics Magazine, 76 (2003), 107118.
Barbara H. Margolius, DinnerDiner Matching Probabilities
S. G. Penrice, Derangements, permanents and Christmas presents, The American Mathematical Monthly 98(1991), 617620.
Index entries for sequences related to card matching


FORMULA

G.f.: sum(coeff(R(x, n, k), x, j)*(t1)^j*(n*kj)!, 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/((kj)!^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 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.


MAPLE

p := (x, k)>k!^2*sum(x^j/((kj)!^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)*(t1)^j*(n*kj)!, j=0..n*k);
for n from 0 to 3 do seq(coeff(f(t, 5, n), t, m), m=0..5*n); od;


MATHEMATICA

p[x_, k_] := k!^2*Sum[ x^j/((kj)!^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]*(t1)^j*(n*kj)!, {j, 0, n*k}]; Table[ Coefficient[ f[t, 5, n], t, m], {n, 0, 3}, {m, 0, 5*n}] // Flatten (* JeanFrançois Alcover, Mar 04 2013, translated from Maple *)


CROSSREFS

Cf. A008290, A059056A059071.
Sequence in context: A239534 A171906 A059070 * A085519 A112815 A242934
Adjacent sequences: A059068 A059069 A059070 * A059072 A059073 A059074


KEYWORD

nonn,tabf,nice


AUTHOR

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


EXTENSIONS

Edited by M. F. Hasler, Sep 20 2015


STATUS

approved



