A057963 Triangle T(n,k) of number of minimal 2-covers of a labeled n-set that cover k points of that set uniquely (k=2,..,n).
1, 3, 3, 6, 12, 7, 10, 30, 35, 15, 15, 60, 105, 90, 31, 21, 105, 245, 315, 217, 63, 28, 168, 490, 840, 868, 504, 127, 36, 252, 882, 1890, 2604, 2268, 1143, 255, 45, 360, 1470, 3780, 6510, 7560, 5715, 2550, 511, 55, 495, 2310, 6930, 14322, 20790, 20955, 14025
Offset: 2
Examples
There are 90=10+30+35+15 minimal 2-covers of a labeled 5-set. Triangle starts: 1; 3, 3; 6, 12, 7; 10, 30, 35, 15; 15, 60, 105, 90, 31; ...
Links
- Robert Israel, Table of n, a(n) for n = 2..10012 (rows 2 to 142, flattened)
- Octavio A. Agustín-Aquino, Archimedes' quadrature of the parabola and minimal covers, arXiv:1602.05279 [math.CO], 2016.
- Eric Weisstein's World of Mathematics, Minimal cover
Programs
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Magma
/* As triangle: */ [[Binomial(n, k)*(2^(k-1)-1): k in [2..n]]: n in [1.. 15]]; // Vincenzo Librandi, Feb 19 2016
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Maple
seq(seq(binomial(n,k)*(2^(k-1)-1),k=2..n), n=2..13); # Robert Israel, Feb 18 2016
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Mathematica
Table[ Binomial[n, k] (2^(k-1)-1), {n, 2, 13}, {k, 2, n}] // Flatten (* Jean-François Alcover, Sep 18 2018, from Maple *) -
PARI
T(n,k) = m=2; binomial(n, k)*stirling(k, m, 2)*(2^m-m-1)^(n-k); \\ Michel Marcus, Feb 18 2016
Formula
Number of minimal m-covers of a labeled n-set that cover k points of that set uniquely is C(n, k)*S(k, m)*(2^m-m-1)^(n-k), where S(k, m) are Stirling numbers of the second kind. Here m=2.
From Robert Israel, Feb 18 2016: (Start)
T(n,k) = C(n,k) * (2^(k-1)-1).
G.f. of triangle: x^2*y^2/((1-x)*(1-x-x*y)*(1-x-2*x*y)). (End)
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