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A077042
Square array read by falling antidiagonals of central polynomial coefficients: largest coefficient in expansion of (1 + x + x^2 + ... + x^(n-1))^k = ((1-x^n)/(1-x))^k, i.e., the coefficient of x^floor(k*(n-1)/2) and of x^ceiling(k*(n-1)/2); also number of compositions of floor(k*(n+1)/2) into exactly k positive integers each no more than n.
22
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, 0, 1, 10, 19, 12, 5, 1, 1, 0, 1, 20, 51, 44, 19, 6, 1, 1, 0, 1, 35, 141, 155, 85, 27, 7, 1, 1, 0, 1, 70, 393, 580, 381, 146, 37, 8, 1, 1, 0, 1, 126, 1107, 2128, 1751, 780, 231, 48, 9, 1, 1, 0, 1, 252, 3139
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
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
Central diagonal is A077045, with A077046 and A077047 either side. Cf. A067059, A270918, A201552.
Cf. A273975.
Sequence in context: A228128 A060959 A342689 * A144903 A356266 A108934
KEYWORD
nonn,tabl
AUTHOR
Henry Bottomley, Oct 22 2002
STATUS
approved