A321126 T(n,k) = max(n + k - 1, n + 1, k + 1), square array read by antidiagonals (n >= 0, k >= 0).

1, 2, 2, 3, 2, 3, 4, 3, 3, 4, 5, 4, 3, 4, 5, 6, 5, 4, 4, 5, 6, 7, 6, 5, 5, 5, 6, 7, 8, 7, 6, 6, 6, 6, 7, 8, 9, 8, 7, 7, 7, 7, 7, 8, 9, 10, 9, 8, 8, 8, 8, 8, 8, 9, 10, 11, 10, 9, 9, 9, 9, 9, 9, 9, 10, 11, 12, 11, 10, 10, 10, 10, 10, 10, 10, 10, 11, 12, 13, 12, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 13

Offset

0,2

Comments

T(n,k) - 1 is the maximum degree of d in the three-variable bracket polynomial <K>(A,B,d) for the two-bridge knot with Conway's notation C(n,k). Hence, T(n,k) is the maximum number of Jordan curves that are obtained by splitting the crossings of such knot diagram.

References

Louis H. Kauffman, Formal Knot Theory, Princeton University Press, 1983.

Links

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened).

Louis H. Kauffman, State models and the Jones polynomial, Topology Vol. 26 (1987), 395-407.

Kelsey Lafferty, The three-variable bracket polynomial for reduced, alternating links, Rose-Hulman Undergraduate Mathematics Journal Vol. 14 (2013), 98-113.

Matthew Overduin, The three-variable bracket polynomial for two-bridge knots, California State University REU, 2013.

Franck Maminirina Ramaharo, Illustration of T(2,2)

Franck Maminirina Ramaharo, Note on sequence A321125 and related ones

Franck Ramaharo, A generating polynomial for the two-bridge knot with Conway's notation C(n,r), arXiv:1902.08989 [math.CO], 2019.

Eric Weisstein's World of Mathematics, Bracket Polynomial

Wikipedia, 2-bridge knot

Wikipedia, Bracket polynomial

Formula

T(n,k) = T(k,n).

T(n,k) = A051125(n+1,k+1) for 0 <= k <= 2, n >= 0, and T(n,k) = A051125(n+1,k+1) + A003983(n-2,k-2) for k >= 3, n >= 3.

T(n,n) = A004280(n+1).

G.f.: (1 - (2*x - x^2)*y + (x - 2*x^2 + x^3)*y^2 + (x^2 - x^3)*y^3)/(((1 - x)*(1 - y))^2).

Example

Square array begins:
    1,  2,  3,  4,  5,  6,  7,  8,  9, 10, ...
    2,  2,  3,  4,  5,  6,  7,  8,  9, 10, ...
    3,  3,  3,  4,  5,  6,  7,  8,  9, 10, ...
    4,  4,  4,  5,  6,  7,  8,  9, 10, 11, ...
    5,  5,  5,  6,  7,  8,  9, 10, 11, 12, ...
    6,  6,  6,  7,  8,  9, 10, 11, 12, 13, ...
    7,  7,  7,  8,  9, 10, 11, 12, 13, 14, ...
    8,  8,  8,  9, 10, 11, 12, 13, 14, 15, ...
    9,  9,  9, 10, 11, 12, 13, 14, 15, 16, ...
   10, 10, 10, 11, 12, 13, 14, 15, 16, 17, ...
  ...

Mathematica

Table[Max[k + 1, n - 1, n - k + 1], {n, 0, 10}, {k, 0, n}] // Flatten

Prog

(Maxima) create_list(max(k + 1, n - 1, n - k + 1), n, 0, 10, k, 0, n);

Crossrefs

T(n,1) = degree of the (n+1)-th row polynomial in A300453.

T(n,k) = degree of the n-th row polynomials in A300454 and A321127, k = 2,n, respectively.

Cf. A321125, A316989.

Cf. A003056, A003983, A051125.

Sequence in context: A034883 A338643 A051125 * A342765 A244580 A131830

Adjacent sequences: A321123 A321124 A321125 * A321127 A321128 A321129

Keyword

nonn,tabl,easy

Author

Franck Maminirina Ramaharo, Nov 21 2018

Status

approved