A340437 Number of sets in the geometry determined by the Hausdorff metric at each location between two sets defined by a complete bipartite graph K(4,n) (with n at least 4) missing three edges, where exactly two of the removed edges are incident to the same vertex in the 4-point set but none of the removed edges are incident to the same vertex in the other set.

3451, 64705, 1068475, 16734721, 256041211, 3875962945, 58387463995, 877549918081, 13175419936891, 197716513589185, 2966344254717115, 44499339824298241, 667519329831422971, 10012994576066466625, 150196351045142283835, 2252955292518089539201, 33794399575693844931451

Offset: 4

Author

Rachel Wofford, Apr 02 2021

Keywords

• easy
• nonn

Comments

Start with a complete bipartite graph K(4,n) with vertex sets A and B where |A| = 4 and |B| is at least 4. We can arrange the points in sets A and B such that h(A,B) = d(a,b) for all a in A and b in B, where h is the Hausdorff metric. The pair [A,B] is a configuration. Then a set C is between A and B at location s if h(A,C) = h(C,B) = h(A,B) and h(A,C) = s. Call a pair ab, where a is in A and b is in B an edge. This sequence provides the number of sets between sets A' and B' at location s in a new configuration [A',B'] obtained from [A,B] by removing three edges, where exactly two of the removed edges are incident to the same point in A but none of the removed edges are incident to the same point in B. So this sequence tells the number of sets at each location on the line segment between A' and B'.

Number of {0,1} 4 X n matrices (with n at least 4) with three fixed zero entries where exactly two zero entries occur in one row and no column has more than one zero entry, with no zero rows or columns.

Take a complete bipartite graph K(4,n) (with n at least 4) having parts A and B where |A| = 4. This sequence gives the number of edge covers of the graph obtained from this K(4,n) graph after removing three edges, where exactly two of the removed edges are incident to the same vertex in A but none of the removed edges are incident to the same vertex in B.

Links

• Steven Schlicker, Roman Vasquez, and Rachel Wofford, Integer Sequences from Configurations in the Hausdorff Metric Geometry via Edge Covers of Bipartite Graphs, J. Int. Seq. (2023) Vol. 26, Art. 23.6.6.
• Index entries for linear recurrences with constant coefficients, signature (26,-196,486,-315).

Crossrefs

Sequences of segments from removing edges from bipartite graphs A335608-A335613, A337416-A337418, A340173-A340175, A340199-A340201, A340897-A340899, A342580, A342796, A342850, A340403-A340405, A340433-A340438, A341551-A341553, A342327-A342328, A343372-A343374, A343800. Polygonal chain sequences A152927, A152928, A152929, A152930, A152931, A152932, A152933, A152934, A152939. Number of {0,1} n X n matrices with no zero rows or columns A048291.

Programs

Mathematica

LinearRecurrence[{26,-196,486,-315},{3451,64705,1068475,16734721},20] (* Harvey P. Dale, Apr 03 2022 *)

Formula

a(n) = 343*15^(n-3) - 264*7^(n-3) + 52*3^(n-3) - 2.

G.f.: x^4*(3451 - 25021*x + 62541*x^2 - 40635*x^3)/(1 - 26*x + 196*x^2 - 486*x^3 + 315*x^4). - Stefano Spezia, Apr 10 2021