A059056 -id:A059056 - cp's OEIS frontend

A059069 Card-matching numbers (Dinner-Diner matching numbers).

1, 9, 8, 6, 0, 1, 4752, 10752, 11776, 7680, 3936, 1024, 384, 0, 16, 17927568, 64105344, 109524960, 117863424, 89474544, 49828608, 21352896, 6718464, 1854576, 279936, 69984, 0, 1296, 248341303296, 1215287525376

Offset: 0

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

This is a triangle of card matching numbers. Two decks each have 4 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..4n). The probability of exactly k matches is T(n,k)/(4n)!.

rows are of length 1,5,9,13,...

			There are 11776 ways of matching exactly 2 cards when there are 2 cards of each kind and 4 kinds of card so T(2,2)=11776.

F. N. David and D. E. Barton, Combinatorial Chance, Hafner, NY, 1962, Ch. 7 and Ch. 12.
S. G. Penrice, Derangements, permanents and Christmas presents, The American Mathematical Monthly 98(1991), 617-620.
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.

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
Index entries for sequences related to card matching

Cf. A008290, A059056-A059071.

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,4,n),t,m),m=0..4*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, 4, n], t, m], {n, 0, 5}, {m, 0, 4*n}] // Flatten (* Jean-François Alcover, Oct 21 2013, after Maple *)

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 (4 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.

A059070 Card-matching numbers (Dinner-Diner matching numbers).

Original entry on oeis.org

1, 44, 45, 20, 10, 0, 1, 13756, 30480, 32365, 21760, 10300, 3568, 970, 160, 40, 0, 1, 6699824, 23123880, 38358540, 40563765, 30573900, 17399178, 7723640, 2729295, 776520, 180100, 33372, 5355, 540, 90, 0, 1

Offset: 0

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

This is a triangle of card matching numbers. A deck has 5 kinds of cards, n of each kind. The deck is shuffled and dealt in to 5 hands with 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..5n). The probability of exactly k matches is T(n,k)/((5n)!/n!^5).

Rows are of length 1,6,11,16,...

			There are 32365 ways of matching exactly 2 cards when there are 2 cards of each kind and 5 kinds of card so T(2,2)=32365.

F. N. David and D. E. Barton, Combinatorial Chance, Hafner, NY, 1962, Ch. 7 and Ch. 12.
S. G. Penrice, Derangements, permanents and Christmas presents, The American Mathematical Monthly 98(1991), 617-620.
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.

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
Index entries for sequences related to card matching

Cf. A008290, A059056-A059071.

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 4 do seq(coeff(f(t,5,n),t,m)/n!^5,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!^5, {n, 0, 4}, {m, 0, 5*n}] // Flatten (* Jean-François Alcover, Oct 21 2013, after Maple *)

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.

cp's OEIS Frontend

A059069 Card-matching numbers (Dinner-Diner matching numbers).

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