A048291 -id:A048291 - cp's OEIS frontend

A283624 Number of {0,1} n X n matrices with no rows or columns in which all entries are the same.

1, 0, 2, 102, 22874, 17633670, 46959933962, 451575174961302, 16271255119687320314, 2253375946574190518740230, 1219041140314101911449662059402, 2601922592659455476330065914740044182, 22040870572750372076278589658097827953983034

Offset: 0

Robert FERREOL, Mar 12 2017

nonn

Every row and column must contain both a 0 and a 1 .

a(n) is the number of relations on n labeled points such that for every point x there exists y,z,t,u such that xRy, zRx, not(xRt), and not(uRx).

			For n=2 the a(2)=2 matrices are
  0 1
  1 0
and
  1 0
  0 1

Cf. A048291.

Diagonal of A283654.

seq(2*sum((-1)^(n+k)*binomial(n,k)*(2^k-1)^n,k=0..n)+2^(n^2)+2*(2^n-2)^n-4*(2^n-1)^n,n=0..10)

Table[If[n==0, 1, 2 Sum[(-1)^(n + k) * Binomial[n, k] * (2^k - 1)^n, {k, 0,n}] + 2^(n^2) + 2*(2^n - 2)^n - 4*(2^n - 1)^n], {n, 0, 12}] (* Indranil Ghosh, Mar 12 2017 *)

for(n=0, 12, print1(2*sum(k=0, n, (-1)^(n + k) * binomial(n, k) * (2^k - 1)^n) + 2^(n^2) + 2*(2^n - 2)^n - 4*(2^n - 1)^n,", ")) \\ Indranil Ghosh, Mar 12 2017

import math
f = math.factorial
def C(n, r): return f(n)//f(r)//f(n - r)
def A(n):
    s=0
    for k in range(0, n+1):
        s+=(-1)**(n + k) * C(n, k) * (2**k -1)**n
    return 2*s + 2**(n**2) + 2*(2**n - 2)**n - 4*(2**n - 1)**n # Indranil Ghosh, Mar 12 2017

a(n) = 2*Sum_{k=0..n} ((-1)^(n+k)*binomial(n,k)*(2^k-1)^n) + 2^(n^2) + 2*(2^n-2)^n - 4*(2^n-1)^n.

a(n) = 2*A048291(n) + 2^(n^2) + 2*(2^n-2)^n - 4*(2^n-1)^n.

a(11)-a(12) from Indranil Ghosh, Mar 12 2017

A297008 Number of edge covers in the complete tripartite graph K_{n,n,n}.

Original entry on oeis.org

4, 2902, 117207580, 268752741193822, 37231937318464496521924, 323097476641999571450657507823382, 178177528846515370073473806783721111760309500, 6274803675843247716007930604166972482973014660984656159102

Offset: 1

Eric W. Weisstein, Dec 23 2017

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..25
Eric Weisstein's World of Mathematics, Complete Tripartite Graph
Eric Weisstein's World of Mathematics, Edge Cover

Cf. A048291, A183109.

b[m_, n_] := Sum[(-1)^j*Binomial[m, j]*If[n == 0, 1, (2^(m - j) - 1)^n], {j, 0, m}];
c[n_, s_] := Sum[Binomial[n, k]*Binomial[n, s - k]*b[k, s - k], {k, Max[0, s - n], Min[n, s]}];
a[n_] := Sum[c[n, 2*n - i]*Sum[(-1)^j*Binomial[i, j]*(2^(2*n - j) - 1)^n, {j, 0, i}], {i, 0, 2 n}];
Array[a, 10] (* Jean-François Alcover, Dec 27 2017, after Andrew Howroyd *)

\\ here b(m,n) is A183109.
b(m, n)={sum(j=0, m, (-1)^j*binomial(m, j)*(2^(m - j) - 1)^n)}
c(n, s)={sum(k=max(0, s-n), min(n, s),binomial(n, k)*binomial(n, s-k)*b(k, s-k))}
a(n)={sum(i=0, 2*n, c(n, 2*n-i)*sum(j=0, i, (-1)^j*binomial(i, j)*(2^(2*n - j) - 1)^n))} \\ Andrew Howroyd, Dec 24 2017

Terms a(4) and beyond from Andrew Howroyd, Dec 24 2017

A370165 Number of labeled loop-graphs covering n vertices without a non-loop edge with loops at both ends.

Original entry on oeis.org

1, 1, 4, 29, 400, 10289, 496548, 45455677, 7983420736, 2716094133313, 1803251169342820, 2348787270663723581, 6024912118926389490448, 30516957491540079828757553, 305811332460677494410532494660, 6071677788061208810793717466942237

Offset: 0

Gus Wiseman, Feb 12 2024

nonn

Number of ways to choose a stable vertex set of a simple graph with n vertices.

			The a(3) = 29 loop-graphs (loops shown as singletons):
  {1,23}   {1,2,3}     {1,2,13,23}
  {2,13}   {1,2,13}    {1,3,12,23}
  {3,12}   {1,2,23}    {2,3,12,13}
  {12,13}  {1,3,12}    {1,12,13,23}
  {12,23}  {1,3,23}    {2,12,13,23}
  {13,23}  {2,3,12}    {3,12,13,23}
           {2,3,13}
           {1,12,13}
           {1,12,23}
           {1,13,23}
           {2,12,13}
           {2,12,23}
           {2,13,23}
           {3,12,13}
           {3,12,23}
           {3,13,23}
           {12,13,23}

Andrew Howroyd, Table of n, a(n) for n = 0..80

Without loops we have A006129, connected A001187.
The non-covering version is A079491.
The unlabeled version is A370166, non-covering A339832.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A000666 counts unlabeled loop-graphs, covering A322700.
A006125 counts labeled loop-graphs (shifted left), covering A322661.
Cf. A048291, A062740, A116539, A320461, A322635.

Table[Length[Select[Subsets[Subsets[Range[n],{1,2}]], Union@@#==Range[n]&&!MatchQ[#, {_,{x_},_,{y_},_,{x_,y_},_}]&]],{n,0,5}]

seq(n)={Vec(serlaplace(sum(k=0, n, exp((2^k-1)*x + O(x*x^n))*2^(k*(k-1)/2)*x^k/k!)))} \\ Andrew Howroyd, Feb 20 2024

Inverse binomial transform of A079491.

E.g.f.: Sum_{k >= 0} exp((2^k-1)*x)*2^(k*(k-1)/2)*x^k/k!. - Andrew Howroyd, Feb 20 2024

A300757 Number of {0,1} n X n matrices with at least one zero row or column.

Original entry on oeis.org

0, 1, 9, 247, 24033, 8556511, 11352479289, 57075144133687, 1103141820795719553, 82892911663509240924031, 24412686635953864779902802489, 28336114545173968837978876913225527, 130112343169670163804689528125616522784993, 2369443670415015818147708710186881118552006568671

Offset: 0

Lee A. Newberg, Mar 12 2018

nonn

Lee A. Newberg, Table of n, a(n) for n = 0..58

Cf. A048291.

a(n) = 2^(n^2) - A048291(n).

A335609 Number of sets (in the Hausdorff metric geometry) at each location between two sets defined by a K(4,n) (with n at least 2) complete bipartite graph missing one edge.

Original entry on oeis.org

26, 896, 18458, 316928, 5049626, 77860736, 1182865178, 17848076288, 268458094106, 4032033838976, 60516655913498, 908002911016448, 13621815273480986, 204339630665964416, 3065181271854043418, 45978326763617681408, 689679155263179402266, 10345217105634885213056

Offset: 2

Steven Schlicker, Jun 15 2020

Number of {0,1} 4 X n matrices (with n at least 2) with one fixed zero entry and no zero rows or columns.

Number of edge covers of a K(4,n) complete bipartite graph (with n at least 2) missing one edge.

			For n = 3, a(2) = 26.

Colin Barker, Table of n, a(n) for n = 2..800
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).

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.

Array[7*15^(# - 1) - 16*7^(# - 1) + 4*3^# - 3 &, 18, 2] (* Michael De Vlieger, Jun 22 2020 *)
LinearRecurrence[{26,-196,486,-315},{26,896,18458,316928},20] (* Harvey P. Dale, Aug 21 2021 *)

Vec(2*x^2*(13 + 110*x + 129*x^2) / ((1 - x)*(1 - 3*x)*(1 - 7*x)*(1 - 15*x)) + O(x^20)) \\ Colin Barker, Jun 23 2020

a(n) = 7*15^(n-1) - 16*7^(n-1) + 4*3^n - 3.
From Colin Barker, Jun 23 2020: (Start)
G.f.: 2*x^2*(13 + 110*x + 129*x^2) / ((1 - x)*(1 - 3*x)*(1 - 7*x)*(1 - 15*x)).
a(n) = 26*a(n-1) - 196*a(n-2) + 486*a(n-3) - 315*a(n-4) for n>5.
(End)

A370166 Number of unlabeled loop-graphs covering n vertices without a non-loop edge with loops at both ends.

Original entry on oeis.org

1, 1, 3, 9, 36, 180, 1313, 14709, 277755, 9304977, 568315345, 63806703305, 13200565313255, 5042653259803433, 3567050969262370941, 4688444463558713135201, 11491940559865490367844649, 52719458629883487816297211441, 454220675869975957947658748125099

Offset: 0

Gus Wiseman, Feb 12 2024

nonn

			Representatives of the a(0) = 1 through a(3) = 9 loop-graphs (loops shown as singletons):
  {}  {{1}}  {{1,2}}      {{1},{2,3}}
             {{1},{2}}    {{1,2},{1,3}}
             {{1},{1,2}}  {{1},{2},{3}}
                          {{1},{2},{1,3}}
                          {{1},{1,2},{1,3}}
                          {{1},{1,2},{2,3}}
                          {{1,2},{1,3},{2,3}}
                          {{1},{2},{1,3},{2,3}}
                          {{1},{1,2},{1,3},{2,3}}

Without loops we have A002494, labeled A006129, connected A001349.
The non-covering version is A339832.
The labeled version is A370165, non-covering A079491 (apparently).
A000666 counts unlabeled loop-graphs, covering A322700.
A006125 counts labeled loop-graphs (shifted left), covering A322661.
Cf. A000085, A001187, A048291, A062740, A116539.

brute[m_]:=First[Sort[Table[Sort[Sort /@ (m/.Rule@@@Table[{(Union@@m)[[i]],p[[i]]},{i,Length[p]}])], {p,Permutations[Range[Length[Union@@m]]]}]]];
Table[Length[Union[brute /@ Select[Subsets[Subsets[Range[n],{1,2}]],Union@@#==Range[n] && !MatchQ[#,{_,{x_},_,{y_},_,{x_,y_},_}]&]]], {n,0,4}]

First differences of A339832 (the non-covering version).

A335610 Number of sets (in the Hausdorff metric geometry) at each location between two sets defined by a K(5,n) (with n at least 2) complete bipartite graph missing one edge.

Original entry on oeis.org

80, 6800, 316928, 11784608, 397551920, 12828154160, 405380093408, 12683426301248, 394943123789840, 12269641330477520, 380755304897252288, 11809363300986469088, 366179512530595589360, 11352903763691009133680, 351960100658771425777568, 10911064386177197162304128

Offset: 2

Steven Schlicker, Jun 15 2020

Number of {0,1} 5 X n matrices (with n at least 2) with one fixed zero entry and no zero rows or columns.

Number of edge covers of a K(5,n) complete bipartite graph (with n at least 2) missing one edge.

			For n = 2, a(2) = 80.

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 (57,-1002,6562,-15381,9765).

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.

Array[15*31^(# - 1) - 43*15^(# - 1) + 46*7^(# - 1) - 22*3^(# - 1) + 4 &, 16, 2] (* Michael De Vlieger, Jun 22 2020 *)

a(n) = 15*31^(n-1) - 43*15^(n-1) + 46*7^(n-1) - 22*3^(n-1) + 4.
From Stefano Spezia, Jul 04 2020: (Start)
G.f.: 16*x^2*(5 + 140*x + 593*x^2 + 522*x^3)/(1 - 57*x + 1002*x^2 - 6562*x^3 + 15381*x^4 - 9765*x^5).
a(n) = 57*a(n-1) - 1002*a(n-2) + 6562*a(n-3) - 15381*a(n-4) + 9765*a(n-5) for n > 6. (End)

A335611 Number of sets (in the Hausdorff metric geometry) at each location between two sets defined by a complete bipartite graph K(6,n) (with n at least 2) missing one edge.

Original entry on oeis.org

242, 49208, 5049626, 397551920, 27839280002, 1845793079528, 119216755050026, 7602793781214560, 481851209165874962, 30446042035976733848, 1920876815510991751226, 121101364739596962016400, 7632056827800217741372322, 480902390923479550619876168

Offset: 2

Steven Schlicker, Jul 16 2020

The Hausdorff metric defines a distance between sets. Using this distance we can define line segments with sets as endpoints. Create two sets from the vertices of the parts A and B of a complete bipartite graph K(6,n) (with n at least 2) missing one edge so that vertices that are connected by edges are the same Euclidean distance apart. This sequence gives the number of sets at each location on the line segment between A and B.
Number of {0,1} 6 X n (with n at least 2) matrices with one fixed zero entry and no zero rows or columns.
Take a complete bipartite graph K(6,n) (with n at least 2). This sequence gives the number of edge covers of the graph obtained from this K(6,n) graph after removing one edge.

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).

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.

a:= proc(n) 31*63^(n-1)-106*31^(n-1)+145*15^(n-1) - 100*7^(n-1)+35*3^(n-1)-5 end proc: seq(a(n), n=2..20);

Vec(2*x^2*(121 + 10084*x + 128086*x^2 + 372324*x^3 + 270585*x^4) / ((1 - x)*(1 - 3*x)*(1 - 7*x)*(1 - 15*x)*(1 - 31*x)*(1 - 63*x)) + O(x^18)) \\ Colin Barker, Jul 17 2020

a(n) = 31*63^(n-1) - 106*31^(n-1) + 145*15^(n-1) - 100*7^(n-1) + 35*3^(n-1) - 5.
From Colin Barker, Jul 17 2020: (Start)
G.f.: 2*x^2*(121 + 10084*x + 128086*x^2 + 372324*x^3 + 270585*x^4) / ((1 - x)*(1 - 3*x)*(1 - 7*x)*(1 - 15*x)*(1 - 31*x)*(1 - 63*x)).
a(n) = 120*a(n-1) - 4593*a(n-2) + 69688*a(n-3) - 428787*a(n-4) + 978768*a(n-5) - 615195*a(n-6) for n>7.
(End)

A335612 Number of sets (in the Hausdorff metric geometry) at each location between two sets defined by a complete bipartite graph K(3,n) (with n at least 3) missing two edges, where the removed edges are incident to the same vertex in the three point part.

Original entry on oeis.org

32, 344, 2792, 20720, 148592, 1050824, 7387832, 51811040, 362965952, 2541627704, 17793992072, 124565738960, 871983556112, 6103955042984, 42727895751512, 299095901612480, 2093673205343072, 14655718119568664, 102590043883482152

Offset: 3

Steven Schlicker, Jul 16 2020

The Hausdorff metric defines a distance between sets. Using this distance we can define line segments with sets as endpoints. Create two sets from the vertices of the parts A and B (with |A| = 3) of a complete bipartite graph K(3,n) (with n at least 3) missing two edges, where the removed edges are incident to the same point in A. Points in the sets A and B that correspond to vertices that are connected by edges are the same Euclidean distance apart. This sequence gives the number of sets at each location on the line segment between the sets A and B.
Number of {0,1} 3 X n matrices (with n at least 3) with two fixed zero entries in the same row and no zero rows or columns.
Take a complete bipartite graph K(3,n) (with n at least 3) having parts A and B where |A| = 3. This sequence gives the number of edge covers of the graph obtained from this K(3,n) graph after removing two edges, where the two removed edges are incident to the same vertex in A.

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 (11,-31,21).

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.

a:= proc(n) 9*7^(n-2)-11*3^(n-2)+2 end proc: seq(a(n), n=3..21);

Vec(8*x^3*(4 - x) / ((1 - x)*(1 - 3*x)*(1 - 7*x)) + O(x^25)) \\ Colin Barker, Jul 17 2020

a(n) = 9*7^(n-2) - 11*3^(n-2) + 2.
From Colin Barker, Jul 17 2020: (Start)
G.f.: 8*x^3*(4 - x) / ((1 - x)*(1 - 3*x)*(1 - 7*x)).
a(n) = 11*a(n-1) - 31*a(n-2) + 21*a(n-3) for n>5.
(End)

A337417 Number of sets (in the Hausdorff metric geometry) at each location between two sets defined by a complete bipartite graph K(6,n) (with n at least 3) missing two edges, where the removed edges are incident to the same vertex in the six point part.

Original entry on oeis.org

16322, 2145368, 183405386, 13292505200, 895227774482, 58252080636488, 3728244541647386, 236702709858383840, 14969004415531532642, 944809197018309879608, 59577646546802243102186, 3755087128633478474841680, 236623057112566045886497202, 14908882367276213189083986728

Offset: 3

Steven Schlicker, Aug 26 2020

The Hausdorff metric defines a distance between sets. Using this distance we can define line segments with sets as endpoints. Create two sets from the vertices of the parts A and B (with |A| = 6) of a complete bipartite graph K(6,n) (with n at least 3) missing two edges, where the removed edges are incident to the same point in A. Points in the sets A and B that correspond to vertices that are connected by edges are the same Euclidean distance apart. 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 (with n at least 3) matrices with two fixed zero entries in the same row and no zero rows or columns.
Take a complete bipartite graph K(6,n) (with n at least 3) 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 two edges, where the two removed edges are incident to the same vertex in A.

S. Schlicker, R. Vasquez, R. Wofford, Integer Sequences from Configurations in the Hausdorff Metric Geometry via Edge Covers of Bipartite Graphs. In preparation.

Index entries for linear recurrences with constant coefficients, signature (120,-4593,69688,-428787,978768,-615195).

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.

a:= proc(n) 961*63^(n-2)-2086*31^(n-2)+1615*15^(n-2) - 580*7^(n-2)+95*3^(n-2) -5 end proc: seq(a(n), n=3..20);

Vec(2*x^3*(8161 + 93364*x + 464086*x^2 + 43284*x^3 + 172305*x^4) / ((1 - x)*(1 - 3*x)*(1 - 7*x)*(1 - 15*x)*(1 - 31*x)*(1 - 63*x)) + O(x^15)) \\ Colin Barker, Nov 19 2020

a(n) = 961*63^(n-2) - 2086*31^(n-2) + 1615*15^(n-2) - 580*7^(n-2) + 95*3^(n-2) - 5.
From Colin Barker, Nov 19 2020: (Start)
G.f.: 2*x^3*(8161 + 93364*x + 464086*x^2 + 43284*x^3 + 172305*x^4) / ((1 - x)*(1 - 3*x)*(1 - 7*x)*(1 - 15*x)*(1 - 31*x)*(1 - 63*x)).
a(n) = 120*a(n-1) - 4593*a(n-2) + 69688*a(n-3) - 428787*a(n-4) + 978768*a(n-5) - 615195*a(n-6) for n>8.
(End)

cp's OEIS Frontend

A283624 Number of {0,1} n X n matrices with no rows or columns in which all entries are the same.

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A297008 Number of edge covers in the complete tripartite graph K_{n,n,n}.

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A370165 Number of labeled loop-graphs covering n vertices without a non-loop edge with loops at both ends.

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A300757 Number of {0,1} n X n matrices with at least one zero row or column.

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A335609 Number of sets (in the Hausdorff metric geometry) at each location between two sets defined by a K(4,n) (with n at least 2) complete bipartite graph missing one edge.

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A370166 Number of unlabeled loop-graphs covering n vertices without a non-loop edge with loops at both ends.

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A335610 Number of sets (in the Hausdorff metric geometry) at each location between two sets defined by a K(5,n) (with n at least 2) complete bipartite graph missing one edge.

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Mathematica

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A335611 Number of sets (in the Hausdorff metric geometry) at each location between two sets defined by a complete bipartite graph K(6,n) (with n at least 2) missing one edge.

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