A079909 Solution to the Dancing School Problem with 4 girls and n+4 boys: f(4,n).

1, 5, 26, 90, 246, 566, 1146, 2106, 3590, 5766, 8826, 12986, 18486, 25590, 34586, 45786, 59526, 76166, 96090, 119706, 147446, 179766, 217146, 260090, 309126, 364806, 427706, 498426, 577590, 665846, 763866, 872346, 992006, 1123590

Offset: 0

Author

Jaap Spies, Jan 28 2003

Keywords

• nonn
• easy

Comments

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. See A079908 for more information.

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

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• 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 for computing A079909.
• Jaap Spies, Sage program for computing the polynomial a(n).
• Jaap Spies, A Bit of Math, The Art of Problem Solving, Jaap Spies Publishers (2019).
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A079908-A079928.

Programs

PARI

Vec(-(x^2+1)*(x^4+10*x^2+1)/(x-1)^5 + O(x^100)) \\ Colin Barker, Jan 04 2015

Formula

a(0)=1, a(1)=5, a(n)=n^4 - 2*n^3 + 9*n^2 - 8*n + 6 (n>=2) found by applying theorem 7.2.1 of Brualdi, Ryser: Combinatorial Matrix Theory.

G.f.: -(x^2+1)*(x^4+10*x^2+1) / (x-1)^5. - Colin Barker, Jan 04 2015

E.g.f.: exp(x)*(6 + 10*x^2 + 4*x^3 + x^4) - 5 - x. - Stefano Spezia, Dec 18 2019

Extensions

More terms from Benoit Cloitre, Jan 29 2003