A079908 Solution to the Dancing School Problem with 3 girls and n+3 boys: f(3,n).

1, 4, 14, 36, 76, 140, 234, 364, 536, 756, 1030, 1364, 1764, 2236, 2786, 3420, 4144, 4964, 5886, 6916, 8060, 9324, 10714, 12236, 13896, 15700, 17654, 19764, 22036, 24476, 27090, 29884, 32864, 36036, 39406, 42980, 46764, 50764, 54986, 59436

Offset: 0

Author

Jaap Spies, Jan 28 2003

Keywords

• nonn
• easy

Comments

The Dancing School Problem: a line of g girls (g>0) with integer heights ranging from m to m+g-1 cm and a line of g+h boys (h>=0) ranging in height from m to m+g+h-1 cm are facing each other in a dancing school (m is the minimal height of both girls and boys).

A girl of height l can choose a boy of her own height or taller with a maximum of l+h cm. We call the number of possible matchings f(g,h).

This problem is equivalent to a rooks problem: The number of possible placings of g non-attacking rooks on a g X g+h chessboard with the restriction i <= j <= i+h for the placement of a rook on square (i,j): f(g,h) = per(B), the permanent of the corresponding (0,1)-matrix B, b(i, j)=1 if and only if i <= j <= i+h

f(g,h) = per(B), the permanent of the (0,1)-matrix B of size g X g+h with b(i,j)=1 if and only if i <= j <= i+h.

For fixed g, f(g,n) is polynomial in n for n >= g-2. See reference.

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..10000
• Lute Kamstra, Problem 29 in Vol. 5/3, No. 1, Mar 2002, p. 96. See also Vol. 5/3, Nos. 3 and 4.
• Jaap Spies, Dancing School Problems, Nieuw Archief voor Wiskunde 5/7 nr. 4, Dec 2006, pp. 283-285.
• Jaap Spies, Dancing School Problems, Permanent solutions of Problem 29.
• Jaap Spies, Sage program to compute f(g,h).
• Jaap Spies, A Bit of Math, The Art of Problem Solving, Jaap Spies Publishers (2019).
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A061989, A079909-A079928.

Cf. Essentially the same as A061989.

Programs

Mathematica

Join[{1},Array[#^3+3*#&,5!,1]] (* Vladimir Joseph Stephan Orlovsky, Jan 08 2011 *)

PARI

concat(1,vector(30,n,n^3+3*n)) \\ Derek Orr, Jun 18 2015

Formula

a(n) = max(1, n^3 + 3*n), found by elementary counting.

G.f.: 1+2*x*(2-x+2*x^2)/(1-x)^4. - R. J. Mathar, Nov 19 2007