A316261 The number of ways to induce a single pinch on a compact 2-manifold with n handles. (Note: The manifold is embedded in Euclidean 2-space, and each pinch partitions it into at most two submanifolds.)

1, 3, 9, 15, 26, 37, 55, 73, 100, 127, 165, 203, 254, 305, 371, 437, 520, 603, 705, 807, 930, 1053, 1199, 1345, 1516, 1687, 1885, 2083, 2310, 2537, 2795, 3053, 3344, 3635, 3961, 4287, 4650, 5013, 5415, 5817, 6260

Offset: 0

Author

Joseph Wheat, Jun 27 2018

Keywords

• nonn
• easy

Comments

The formula for this sequence can be derived by separating the conformed manifolds into three sets. The first set consists of those conformations where the handles of the manifold are pinched at the boundary, the second set have two or more handles pinched at the interior of the manifold, and the third set are pinched at the boundary and may or may not have handles drawn into this pinch. The order of the first set is n, the order of the second is n - 1, and the order of the third set is given by the following series: (Sum_{k mod 2 = 0..n} (k/2)*(n - k + 1) + (2*(n - k) + (-1)^(n - k) + 3)/4) + (Sum_{j mod 2 = 1..n} ((j + 1)/2)*(n - j + 1)). These can then be combined into a single expression, Sum_{i = 0..n} ((2*i + (-1)^(i + 1) + 1)/4)*(n - i + 1) + ((2*(n - i) + (-1)^(n - i) + 3)/4)*(((-1)^i + 1)/2). The i in this series can be thought of as the number of handles drawn into the central pinch. If one factors out the expressions in the series and simplifies each term individually, the resulting functions can then be combined into a single formula. However, when we add 2n - 1 to this we find that for n = 0 the formula also equals zero. This cannot be, because there is one way to pinch a compact 2-manifold with 0 handles. Therefore, ((-1)^(2^n - 1) + 1)/2 is added as a corrective term for this one case.

Examples

			For a visual example see links.
		

References

• Jonathan L. Gross, Jay Yellen, and Ping Zhang, The Handbook of Graph Theory (Second Edition), CRC Press, 2013, pp. 730-806.
• Ana Claudia Nabarro, Juan J. Nuño-Ballesteros, Raúl Oset Sinha, Maria Aparecida Soares Ruas, Contemporary Mathematics: Real and Complex Singularities, American Mathematical Soc., 2014, pp. 50-51.

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Jonathan L. Gross, Jay Yellen, and Ping Zhang, The Handbook of Graph Theory (Second Edition)
• Allen Hatcher, Algebraic Topology (Ch. 0)
• Ana Claudia Nabarro, Juan J. Nuño-Ballesteros, Raúl Oset Sinha, Maria Aparecida Soares Ruas, Contemporary Mathematics: Real and Complex Singularities
• Pseudomanifold. The Encyclopedia of Mathematics
• Joseph Wheat, Visual Example of n = 3
• Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Crossrefs

Cf. A087811.

Programs

Mathematica

a[n_] := (2 n^3 + 12 n^2 + 73 n + 3 (n + 2)*(-1)^n - 6)/24 + ((-1)^(2^n - 1) + 1)/2; Array[a, 50, 0] (* or *)
CoefficientList[ Series[(x^6 + x^5 - 2x^4 - 2x^3 + 2x^2 + x + 1)/((x - 1)^4 (x + 1)^2), {x, 0, 50}], x] (* Robert G. Wilson v, Jul 23 2018 *)

PARI

Vec((1 + x + 2*x^2 - 2*x^3 - 2*x^4 + x^5 + x^6) / ((1 - x)^4*(1 + x)^2) + O(x^50)) \\ Colin Barker, Jul 05 2018

Formula

a(n) = (2*n^3 + 12*n^2 + 73*n + 3*(n + 2)*(-1)^n - 6)/24 + ((-1)^(2^n - 1) + 1)/2.

From Colin Barker, Jul 05 2018: (Start)

G.f.: (1 + x + 2*x^2 - 2*x^3 - 2*x^4 + x^5 + x^6) / ((1 - x)^4*(1 + x)^2).

a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6) for n>6.

(End)