A343373 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 removed edges are incident to the same vertex in the 4-point set and exactly two removed edges are incident to the same vertex in the other set.

3112, 58984, 978064, 15345952, 234980152, 3558436504, 53613281824, 805858151632, 12099490097992, 181573692295624, 2724174818398384, 40866608458275712, 613027030583891032, 9195600786027620344

Offset

4,1

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 removed edges are incident to the same point in A and exactly two 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 exactly two zero entries occur in one column, 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 removed edges are incident to the same vertex in A and exactly two removed edges are incident to the same vertex in B.

Links

Table of n, a(n) for n=4..17.

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).

Formula

a(n) = 21*15^(n-2) - 36*7^(n-2) + 17*3^(n-2) - 2.

G.f.: 8*x^4*(389 - 2741*x + 6804*x^2 - 4410*x^3)/(1 - 26*x + 196*x^2 - 486*x^3 + 315*x^4). - Stefano Spezia, Apr 13 2021

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.

Sequence in context: A187593 A230923 A137839 * A238513 A357957 A269324

Adjacent sequences: A343370 A343371 A343372 * A343374 A343375 A343376

Keyword

easy,nonn

Author

Steven Schlicker, Apr 12 2021

Status

approved