A342328 Number of sets in the geometry determined by the Hausdorff metric at each location between two sets defined by a complete bipartite graph K(6,n) (with n at least 4) missing three edges, where all three removed edges are incident to different vertices in the 6-point set but exactly two removed edges are incident to the same vertex in the other set.

1068475, 89633839, 6458329435, 433976684431, 28211055010555, 1804746233554159, 114556965257054875, 7243790885015626831, 457188176014823960635, 28828588756092946562479, 1816999192589895468925915, 114495695622871975031439631

Offset

4,1

Comments

Start with a complete bipartite graph K(6,n) with vertex sets A and B where |A| = 6 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 all three removed edges are incident to different points in A but 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} 6 X n matrices (with n at least 4) with three fixed zero entries where exactly two zero entries occur in one column and no row has more than one zero entry, with no zero rows or columns.

Take a complete bipartite graph K(6,n) (with n at least 4) having parts A and B where |A| = 6. This sequence gives the number of edge covers of the graph obtained from this K(6,n) graph after removing three edges, where all three removed edges are incident to different vertices in A but exactly two removed edges are incident to the same vertex in B.

Links

Table of n, a(n) for n=4..15.

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 (120,-4593,69688,-428787,978768,-615195).

Formula

a(n) = 465*63^(n-2) - 982*31^(n-2) + 807*15^(n-2) - 316*7^(n-2) + 56*3^(n-2) - 3.

Mathematica

Array[465*63^(# - 2) - 982*31^(# - 2) + 807*15^(# - 2) - 316*7^(# - 2) + 56*3^(# - 2) - 3 &, 12, 4] (* Michael De Vlieger, Mar 19 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: A081638 A193054 A160687 * A070042 A258061 A187125

Adjacent sequences: A342325 A342326 A342327 * A342329 A342330 A342331

Keyword

easy,nonn

Author

Steven Schlicker, Mar 08 2021

Status

approved