OFFSET
0,13
COMMENTS
From Michel Marcus, Dec 01 2012: (Start)
A pair of numbers written in base n are said to be comparable if all digits of the first number are at least as big as the corresponding digit of the second number, or vice versa. Otherwise, this pair will be defined as uncomparable. A set of pairwise uncomparable integers will be called anti-hierarchic.
T(n,k) is the size of the maximal anti-hierarchic set of integers written with k digits in base n.
For example, for base n=2 and k=4 digits:
- 0 (0000) and 15 (1111) are comparable, while 6 (0110) and 9 (1001) are uncomparable,
- the maximal antihierarchic set is {3 (0011), 5 (0101), 6 (0110), 9 (1001), 10 (1010), 12 (1100)} with 6 elements that are all pairwise uncomparable. (End)
LINKS
G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Denis Bouyssou, Thierry Marchant, Marc Pirlot, The size of the largest antichains in products of linear orders, arXiv:1903.07569 [math.CO], 2019.
J. W. Sander, On maximal antihierarchic sets of integers, Discrete Mathematics, Volume 113, Issues 1-3, 5 April 1993, Pages 179-189.
FORMULA
By the central limit theorem, T(n,k) is roughly n^(k-1)*sqrt(6/(Pi*k)).
T(n,k) = Sum{j=0,h/n} (-1)^j*binomial(k,j)*binomial(k-1+h-n*j,k-1) with h=floor(k*(n-1)/2), k>0. - Michel Marcus, Dec 01 2012
EXAMPLE
Rows of square array start:
1, 0, 0, 0, 0, 0, 0, ...
1, 1, 1, 1, 1, 1, 1, ...
1, 1, 2, 3, 6, 10, 20, ...
1, 1, 3, 7, 19, 51, 141, ...
1, 1, 4, 12, 44, 155, 580, ...
1, 1, 5, 19, 85, 381, 1751, ...
...
Read by antidiagonals:
1;
0, 1;
0, 1, 1;
0, 1, 1, 1;
0, 1, 2, 1, 1;
0, 1, 3, 3, 1, 1;
0, 1, 6, 7, 4, 1, 1;
...
MATHEMATICA
t[n_, k_] := Max[ CoefficientList[ Series[ ((1-x^n) / (1-x))^k, {x, 0, k*(n-1)}], x]]; t[0, 0] = 1; t[0, _] = 0; Flatten[ Table[ t[n-k, k], {n, 0, 12}, {k, n, 0, -1}]] (* Jean-François Alcover, Nov 04 2011 *)
PROG
(PARI) T(n, k)=if(n<1 || k<1, k==0, vecmax(Vec(((1-x^n)/(1-x))^k)))
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Henry Bottomley, Oct 22 2002
STATUS
approved