A321127 -id:A321127 - cp's OEIS frontend

A321125 T(n,k) = b(n+k) - (2b(n)b(k) + 1)b(nk) + b(n) + b(k) + 1, where b(n) = A154272(n+1), square array read by antidiagonals (n >= 0, k >= 0).

1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1

Offset: 0

Franck Maminirina Ramaharo, Nov 24 2018

Let (A,B,d) denote the three-variable bracket polynomial for the two-bridge knot with Conway's notation C(n,k). Then T(n,k) is the leading coefficient of the reduced polynomial x*(1,1,x). In Kauffman's language, T(n,k) is the number of states of the two-bridge knot C(n,k) corresponding to the maximum number of Jordan curves.

			Square array begins:
  1, 1, 1, 1, 1, 1, ...
  1, 2, 1, 1, 1, 1, ...
  1, 1, 3, 2, 2, 2, ...
  1, 1, 2, 1, 1, 1, ...
  1, 1, 2, 1, 1, 1, ...
  1, 1, 2, 1, 1, 1, ...
  ...

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

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 this sequence 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

Cf. A300453, A300454, A316989, A321126, A321127.

b[n_] = If[n == 0 || n == 2, 1, 0];
T[n_, k_] = b[n + k] - (2*b[n]*b[k] + 1)*b[n*k] + b[n] + b[k] + 1;
Table[T[k, n - k], {n, 0, 12}, {k, 0, n}] // Flatten

b(n) := if n = 0 or n = 2 then 1 else 0$ /* A154272(n+1) */
T(n, k) := b(n + k) - (2*b(n)*b(k) + 1)*b(n*k) + b(n) + b(k) + 1$
create_list(T(k, n - k), n, 0, 12, k, 0, n);

T(n,0) = T(0,n) = 1, and T(n,k) = b(n+k) - b(n)*b(k) - b(n*k) + c(n)*c(k) for n >= 1, k >= 1, where b(n) = A154272(n+1) and c(n) = A294619(n).
T(n,1) = A300453(n+1,A321126(n,1)).
T(n,2) = A300454(n,A321126(n,2)).
T(n,n) = A321127(n,A004280(n+1)).
G.f.: (1 + (x - x^2)*y - (x - 3*x^2 + x^3)*y^2 - x^2*y^3)/((1 - x)*(1 - y)).
E.g.f.: ((x^2 + 2*exp(x))*exp(y) - x^2 + (2*x - x^2)*y - (1 + x - exp(x))*y^2)/2.

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

Original entry on oeis.org

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

Franck Maminirina Ramaharo, Nov 21 2018

T(n,k) - 1 is the maximum degree of d in the three-variable bracket polynomial (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.

			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, ...
  ...

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

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

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.

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

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

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).

cp's OEIS Frontend

A321125 T(n,k) = b(n+k) - (2b(n)b(k) + 1)b(nk) + b(n) + b(k) + 1, where b(n) = A154272(n+1), square array read by antidiagonals (n >= 0, k >= 0).

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

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cp's OEIS Frontend

A321125 T(n,k) = b(n+k) - (2*b(n)*b(k) + 1)*b(n*k) + b(n) + b(k) + 1, where b(n) = A154272(n+1), square array read by antidiagonals (n >= 0, k >= 0).

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Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Maxima

Formula

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Maxima

Formula

A321125 T(n,k) = b(n+k) - (2b(n)b(k) + 1)b(nk) + b(n) + b(k) + 1, where b(n) = A154272(n+1), square array read by antidiagonals (n >= 0, k >= 0).