Travis Hoppe - cp's OEIS frontend

A355616 a(n) is the number of distinct lengths between consecutive points of the Farey sequence of order n.

1, 1, 2, 3, 5, 6, 9, 11, 14, 15, 21, 23, 29, 31, 34, 38, 48, 49, 59, 63, 67, 71, 83, 86, 97, 100, 110, 115, 132, 133, 150, 158, 165, 169, 182, 187, 208, 213, 222, 228, 252, 254, 280, 287, 297, 304, 331, 337, 362, 367, 379, 387, 418, 423, 437, 450, 464, 472, 509, 513, 548, 556, 573, 589, 608, 611, 652, 665, 681, 685

Offset: 1

Travis Hoppe, Jul 09 2022

nonn

The Farey sequence of order n (row n of A006842/A006843) is the set of points x/y on the unit line where 1 <= y <= n and 0 <= x <= y.

			For n=5, the Farey sequence (completely reduced fractions) is [0/1, 1/5, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5, 1/1]. The distinct lengths between consecutive points are {1/5, 1/20, 1/12, 1/15, 1/10} so a(5) = 5.

Cf. A006842/A006843 (Farey sequences).

Cf. A005728 (number of distinct points).

a[n_] := FareySequence[n] // Differences // Union // Length;
Table[a[n], {n, 1, 70}] (* Jean-François Alcover, Jul 16 2022 *)

vp(n) = my(list = List()); for (k=1, n, for (i=0, k, listput(list, i/k))); vecsort(list,,8);
a(n) = my(v=vp(n)); #Set(vector(#v-1, k, abs(v[k+1]-v[k]))); \\ Michel Marcus, Jul 10 2022

from fractions import Fraction
from itertools import chain
def compute(n):
    marks = [[(a, b) for a in range(0, b + 1)] for b in range(1, n + 1)]
    marks = sorted(set([Fraction(a, b) for a, b in chain(*marks)]))
    dist = [(y - x) for x, y in zip(marks, marks[1:])]
    return len(set(dist))

A245882 Number of distinct Laplacian polynomials among all connected graphs on n nodes.

Original entry on oeis.org

1, 1, 2, 6, 21, 110, 793, 10251, 239307, 10985229

Offset: 1

Travis Hoppe and Anna Petrone, Aug 05 2014

The Laplacian polynomial is the characteristic polynomial of the Laplacian matrix, L = A-D where A is the adjacency matrix and D the degree matrix.

Travis Hoppe and Anna Petrone, Encyclopedia of Finite Graphs
T. Hoppe and A. Petrone, Integer sequence discovery from small graphs, arXiv preprint arXiv:1408.3644, 2014
Eric Weisstein's World of Mathematics, Laplacian Polynomial

A245883 Number of distinct chromatic polynomials among all connected graphs on n nodes.

Original entry on oeis.org

1, 1, 2, 5, 14, 50, 231, 1650, 21121, 584432

Offset: 1

Travis Hoppe and Anna Petrone, Aug 05 2014

A stable partition of a graph is a set partition of the vertices where no edge has both ends in the same block. The chromatic polynomial is given by chi_G(x) = Sum_p (x)k, where the sum is over all stable partitions of G, k is the length (number of blocks) of p, and (x)_k is the falling factorial x(x-1)(x-2)...(x-k+1). - _Gus Wiseman, Nov 24 2018

			From _Gus Wiseman_, Nov 24 2018: (Start)
The a(4) = 5 chromatic polynomials:
  -6x + 11x^2 - 6x^3 + x^4
  -4x +  8x^2 - 5x^3 + x^4
  -2x +  5x^2 - 4x^3 + x^4
  -3x +  6x^2 - 4x^3 + x^4
   -x +  3x^2 - 3x^3 + x^4
(End)

Travis Hoppe and Anna Petrone, Encyclopedia of Finite Graphs
T. Hoppe and A. Petrone, Integer sequence discovery from small graphs, arXiv preprint arXiv:1408.3644 [math.CO], 2014.
Eric Weisstein's World of Mathematics, Chromatic Polynomial

Cf. A229048 (simple graphs, including disconnected ones, with unique chromatic polynomials).

Cf. A001187, A001349, A006125, A125702, A229048, A240936, A245883, A277203, A321911, A322011.

spsu[,{}]:={{}};spsu[foo,set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@spsu[Select[foo,Complement[#,Complement[set,s]]=={}&],Complement[set,s]]]/@Cases[foo,{i,_}];
falling[x_,k_]:=Product[(x-i),{i,0,k-1}];
chromPoly[g_]:=Expand[Sum[falling[x,Length[stn]],{stn,spsu[Select[Subsets[Union@@g],Select[DeleteCases[g,{_}],Function[ed,Complement[ed,#]=={}]]=={}&],Union@@g]}]];
simpleSpans[n_]:=simpleSpans[n]=If[n==0,{{}},Union@@Table[If[#=={},Union[ine,{{n}}],Union[Complement[ine,List/@#],{#,n}&/@#]]&/@Subsets[Range[n-1]],{ine,simpleSpans[n-1]}]];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Union[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[Length[Union[chromPoly/@Select[simpleSpans[n],Length[csm[#]]==1&]]],{n,5}] (* Gus Wiseman, Nov 24 2018 *)

A245881 Number of distinct Tutte polynomials among all connected graphs on n nodes.

Original entry on oeis.org

1, 1, 2, 5, 16, 73, 532, 7245, 192605, 9717156

Offset: 1

Travis Hoppe and Anna Petrone, Aug 05 2014

Travis Hoppe and Anna Petrone, Encyclopedia of Finite Graphs
T. Hoppe and A. Petrone, Integer sequence discovery from small graphs, arXiv preprint arXiv:1408.3644 [math.CO], 2014.
T. Hoppe and A. Petrone, Integer sequence discovery from small graphs, Discr. Appl. Math. 201 (2016) 172-181.
Eric Weisstein's World of Mathematics, Tutte Polynomial

Cf. A243049 (simple graphs, including disconnected ones, with unique Tutte polynomials).

Terms corrected by Eric M. Schmidt, Mar 20 2015

A245880 Number of distinct characteristic polynomials among all connected graphs on n nodes.

Original entry on oeis.org

1, 1, 2, 6, 21, 111, 821, 10423, 236064, 10375796

Offset: 1

Travis Hoppe and Anna Petrone, Aug 05 2014

Travis Hoppe and Anna Petrone, Encyclopedia of Finite Graphs
T. Hoppe and A. Petrone, Integer sequence discovery from small graphs, arXiv preprint arXiv:1408.3644 [math.CO], 2014.
T. Hoppe and A. Petrone, Integer sequence discovery from small graphs, Discr. Appl. Math. 201 (2016) 172-181.
Eric Weisstein's World of Mathematics, Characteristic Polynomial

Cf. A082104 (simple graphs, including disconnected ones, with unique characteristic polynomials).

Terms corrected by Eric M. Schmidt, Mar 20 2015

A245879 Number of distinct fractional chromatic numbers among all connected graphs on n nodes.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 17, 29, 50

Offset: 1

Travis Hoppe and Anna Petrone, Aug 05 2014

Travis Hoppe and Anna Petrone, Encyclopedia of Finite Graphs
T. Hoppe and A. Petrone, Integer sequence discovery from small graphs, arXiv preprint arXiv:1408.3644, 2014
Eric Weisstein's World of Mathematics, Fractional Chromatic Number

Cf. A243251 (fractional chromatic gap).

A243800 Number of simple connected graphs on n nodes whose maximum size of an independent edge set is equal to 2.

Original entry on oeis.org

0, 0, 0, 5, 20, 16, 22, 29, 37, 46

Offset: 1

Travis Hoppe and Anna Petrone, Jul 15 2014

An independent edge set is a subset of the edges such that no two edges in the subset share a vertex. A maximum independent edge set is an independent edge set containing the largest possible number of edges among all independent edge sets.

The maximum size of an independent edge set is also called the matching number of the graph. - Andrew Howroyd, Sep 05 2019

Travis Hoppe and Anna Petrone, Encyclopedia of Finite Graphs
T. Hoppe and A. Petrone, Integer sequence discovery from small graphs, arXiv preprint arXiv:1408.3644, 2014
Eric Weisstein's World of Mathematics, Matching Number
Eric Weisstein's World of Mathematics, Maximum Independent Edge Set

Column k=2 of A325304.

Cf. A243801 (max size of independent edge set=3).

A243801 Number of simple connected graphs on n nodes whose maximum size of an independent edge set is equal to 3.

Original entry on oeis.org

0, 0, 0, 0, 0, 95, 830, 790, 1479, 2625

Offset: 1

Travis Hoppe and Anna Petrone, Jul 15 2014

An independent edge set is a subset of the edges such that no two edges in the subset share a vertex. A maximum independent edge set an independent edge set containing the largest possible number of edges among all independent edge sets.

The maximum size of an independent edge set is also called the matching number of the graph. - Andrew Howroyd, Sep 05 2019

Travis Hoppe and Anna Petrone, Encyclopedia of Finite Graphs
T. Hoppe and A. Petrone, Integer sequence discovery from small graphs, arXiv preprint arXiv:1408.3644, 2014
Eric Weisstein's World of Mathematics, Matching Number
Eric Weisstein's World of Mathematics, Maximum Independent Edge Set

Column k=2 of A325304.

Cf. A243800 (max size of independent edge set=2).

A244427 Number of unlabeled, connected graphs on n vertices with no subgraph isomorphic to a bull-graph.

Original entry on oeis.org

1, 1, 2, 6, 9, 26, 80, 340, 1690, 11432, 101142, 1228608, 20329150

Offset: 1

Travis Hoppe and Anna Petrone, Jun 27 2014

Travis Hoppe and Anna Petrone, Encyclopedia of Finite Graphs
F. Hüffner, tinygraph, software for generating integer sequences based on graph properties, version 6c1dbe4
Eric Weisstein's World of Mathematics, Bull Graph

Cf. A079575 (no induced bull subgraph).

a(11)-a(13) added using tinygraph by Falk Hüffner, Jul 01 2018

A243799 Number of connected graphs with n nodes that are chordal and are open-bowtie free.

Original entry on oeis.org

1, 1, 2, 5, 6, 13, 25, 58, 130, 316, 769, 1962, 5052, 13342, 35629, 96671

Offset: 1

Travis Hoppe and Anna Petrone, Jun 27 2014

The open bowtie graph is also known as a cricket. - Falk Hüffner, Jul 01 2018

Travis Hoppe and Anna Petrone, Encyclopedia of Finite Graphs
T. Hoppe and A. Petrone, Integer sequence discovery from small graphs, arXiv preprint arXiv:1408.3644 [math.CO], 2014.
F. Hüffner, tinygraph, Software for generating integer sequences based on graph properties, version 6c1dbe4.

Cf. A048192 (chordal graphs), A242791 (open-bowtie free graphs).

Definition corrected (connected only) by Falk Hüffner, Jul 01 2018

a(11)-a(16) added using tinygraph by Falk Hüffner, Jul 01 2018

cp's OEIS Frontend

User: Travis Hoppe

A355616 a(n) is the number of distinct lengths between consecutive points of the Farey sequence of order n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Python

A245882 Number of distinct Laplacian polynomials among all connected graphs on n nodes.

Views

Author

Keywords

Comments

Links

A245883 Number of distinct chromatic polynomials among all connected graphs on n nodes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A245881 Number of distinct Tutte polynomials among all connected graphs on n nodes.

Views

Author

Keywords

Links

Crossrefs

Extensions

A245880 Number of distinct characteristic polynomials among all connected graphs on n nodes.

Views

Author

Keywords

Links

Crossrefs

Extensions

A245879 Number of distinct fractional chromatic numbers among all connected graphs on n nodes.

Views

Author

Keywords

Links

Crossrefs

A243800 Number of simple connected graphs on n nodes whose maximum size of an independent edge set is equal to 2.

Views

Author

Keywords

Comments

Links

Crossrefs

A243801 Number of simple connected graphs on n nodes whose maximum size of an independent edge set is equal to 3.

Views

Author

Keywords

Comments

Links

Crossrefs

A244427 Number of unlabeled, connected graphs on n vertices with no subgraph isomorphic to a bull-graph.

Views

Author

Keywords

Links

Crossrefs

Extensions

A243799 Number of connected graphs with n nodes that are chordal and are open-bowtie free.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions