OFFSET

0,2

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, 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).

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

MATHEMATICA

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

PROG

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

CROSSREFS

KEYWORD

nonn,easy

AUTHOR

Jaap Spies, Jan 28 2003

STATUS

approved