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URL: https://oeis.org/A381451

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A381451
Triangle read by rows: T(n,k) is the clique covering number of the Johnson graph J(n, k), n >= 2, 0 < k < n.
0
1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 6, 4, 1, 1, 5, 9, 9, 5, 1, 1, 6, 12, 14, 12, 6, 1, 1, 7, 16, 25, 25, 16, 7, 1, 1, 8, 20, 40, 46, 40, 20, 8, 1, 1, 9, 25, 56
OFFSET
2,5
COMMENTS
T(2*k, k) = C(k) = A000108(k), the k-th Catalan number, for k = 1, 2, 4, 6, 8, 16; whether this holds for other values of k is an open question.
LINKS
Søren Fuglede Jørgensen, On the clique covering numbers of Johnson graphs, Des. Codes Cryptogr. (2025); arXiv:2502.15019 [math.CO], 2025.
Eric Weisstein's World of Mathematics, Clique Covering Number.
Eric Weisstein's World of Mathematics, Johnson Graph.
Wikipedia, Johnson graph.
FORMULA
T(n, k) = T(n, n - k).
T(n, 1) = 1.
T(n, 2) = n - 2.
T(n, 3) = A002620(n-1), for n >= 6.
T(n, k) <= T(n - 1, k - 1) + T(n - 1, k).
EXAMPLE
Triangle begins:
n\k 1 2 3 4 5 6 7 8 9 10
2: 1
3: 1 1
4: 1 2 1
5: 1 3 3 1
6: 1 4 6 4 1
7: 1 5 9 9 5 1
8: 1 6 12 14 12 6 1
9: 1 7 16 25 25 16 7 1
10: 1 8 20 40 46 40 20 8 1
11: 1 9 25 56 ? ? 56 25 9 1
...
CROSSREFS
Cf. A002620 (column 3).
Sequence in context: A095142 A180171 A140822 * A212954 A299807 A089239
KEYWORD
nonn,tabl,hard,more
AUTHOR
STATUS
approved