A368600 -id:A368600 - cp's OEIS frontend

A369143 Number of labeled simple graphs with n edges and n vertices such that it is not possible to choose a different vertex from each edge (non-choosable).

0, 0, 0, 0, 0, 30, 1335, 47460, 1651230, 59636640, 2284113762, 93498908580, 4099070635935, 192365988161490, 9646654985111430, 515736895712230192, 29321225548502776980, 1768139644819077541440, 112805126206185257070660, 7595507651522103787077270, 538504704005397535690160274

Offset: 0

Gus Wiseman, Jan 21 2024

nonn

			The term a(5) = 30 counts all permutations of the graph {{1,2},{1,3},{1,4},{2,3},{2,4}}.

Andrew Howroyd, Table of n, a(n) for n = 0..200

The version without the choice condition is A116508, covering A367863.
The complement is A137916.
Allowing any number of edges gives A367867, covering A367868.
The version with loops is A368596, covering A368730, unlabeled A368835.
For set-systems we have A368600, for any number of edges A367903.
The covering case is A369144.
A006125 counts simple graphs, unlabeled A000088.
A058891 counts set-systems (without singletons A016031), unlabeled A000612.
Cf. A000085, A057500, A062740, A129271, A133686, A333331, A367769, A367869, A367901, A367907.

Table[Length[Select[Subsets[Subsets[Range[n],{2}], {n}],Length[Select[Tuples[#],UnsameQ@@#&]]==0&]],{n,0,5}]

a(n) = A116508(n) - A137916(n). - Andrew Howroyd, Feb 02 2024

a(8) onwards from Andrew Howroyd, Feb 02 2024

A369144 Number of labeled simple graphs with n edges covering n vertices such that it is not possible to choose a different vertex from each edge (non-choosable).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 90, 4935, 200970, 7636860, 291089610, 11459170800, 471932476290, 20447369179380, 933942958593645, 44981469288560805, 2282792616992648670, 121924195590795244920, 6843305987751060036720, 403003907531795513467260, 24861219342100679072572470

Offset: 0

Gus Wiseman, Jan 21 2024

nonn

			The term a(6) = 90 counts all permutations of the (non-connected) graph {{1,2},{1,3},{1,4},{2,3},{2,4},{5,6}}.

Andrew Howroyd, Table of n, a(n) for n = 0..200

The covering complement is counted by A137916.
Without the choice condition we have A367863, covering case of A116508.
Allowing any number of edges gives A367868, covering case of A367867.
With loops we have A368730, covering case of A368596, unlabeled A368835.
This is the covering case of A369143.
A003465 counts covering set-systems, unlabeled A055621.
A006125 counts simple graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
A058891 counts set-systems, unlabeled A000612.
A322661 counts covering loop-graphs, connected A062740.
Cf. A000085, A057500, A129271, A133686, A333331, A367869, A367901, A367903, A367907, A368600.

Table[Length[Select[Subsets[Subsets[Range[n],{2}], {n}],Union@@#==Range[n]&&Length[Select[Tuples[#], UnsameQ@@#&]]==0&]],{n,0,6}]

a(n) = A367863(n) - A137916(n). - Andrew Howroyd, Feb 02 2024

a(8) onwards from Andrew Howroyd, Feb 02 2024

A369201 Number of unlabeled simple graphs with n vertices and n edges such that it is not possible to choose a different vertex from each edge (non-choosable).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 7, 30, 124, 507, 2036, 8216, 33515, 138557, 583040, 2503093, 10985364, 49361893, 227342301, 1073896332, 5204340846, 25874724616, 131937166616, 689653979583, 3693193801069, 20247844510508, 113564665880028, 651138092719098, 3813739129140469

Offset: 0

Gus Wiseman, Jan 22 2024

nonn

These are graphs with n vertices and n edges having at least two cycles in the same component.

			The a(0) = 0 through a(6) = 7 simple graphs:
  .  .  .  .  .  {{12}{13}{14}{23}{24}}  {{12}{13}{14}{15}{23}{24}}
                                         {{12}{13}{14}{15}{23}{45}}
                                         {{12}{13}{14}{23}{24}{34}}
                                         {{12}{13}{14}{23}{24}{35}}
                                         {{12}{13}{14}{23}{24}{56}}
                                         {{12}{13}{14}{23}{25}{45}}
                                         {{12}{13}{14}{25}{35}{45}}

Andrew Howroyd, Table of n, a(n) for n = 0..50
Gus Wiseman, The a(6) = 7 unlabeled non-choosable graphs with 6 vertices and 6 edges.

Without the choice condition we have A001434, covering A006649.
The labeled version without choice is A116508, covering A367863, A367862.
The complement is counted by A137917, labeled A137916.
For any number of edges we have A140637, complement A134964.
For labeled set-systems we have A368600.
The case with loops is A368835, labeled A368596.
The labeled version is A369143, covering A369144.
A006129 counts covering graphs, unlabeled A002494.
A007716 counts unlabeled multiset partitions, connected A007718.
A054548 counts graphs covering n vertices with k edges, with loops A369199.
A129271 counts connected choosable simple graphs, unlabeled A005703.
Cf. A000088, A000612, A014068, A053530, A133686, A140638, A368601, A369141, A369146.

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],{2}],{n}],Select[Tuples[#],UnsameQ@@#&]=={}&]]],{n,0,5}]

a(n) = A001434(n) - A137917(n).

a(25) onwards from Andrew Howroyd, Feb 02 2024

A361718 Triangular array read by rows. T(n,k) is the number of labeled directed acyclic graphs on [n] with exactly k nodes of indegree 0.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 15, 9, 1, 0, 316, 198, 28, 1, 0, 16885, 10710, 1610, 75, 1, 0, 2174586, 1384335, 211820, 10575, 186, 1, 0, 654313415, 416990763, 64144675, 3268125, 61845, 441, 1, 0, 450179768312, 286992935964, 44218682312, 2266772550, 43832264, 336924, 1016, 1

Offset: 0

Geoffrey Critzer, Apr 02 2023

Also the number of sets of n nonempty subsets of {1..n}, k of which are singletons, such that there is only one way to choose a different element from each. For example, row n = 3 counts the following set-systems:
  {{1},{1,2},{1,3}}    {{1},{2},{1,3}}    {{1},{2},{3}}
  {{1},{1,2},{2,3}}    {{1},{2},{2,3}}
  {{1},{1,3},{2,3}}    {{1},{3},{1,2}}
  {{2},{1,2},{1,3}}    {{1},{3},{2,3}}
  {{2},{1,2},{2,3}}    {{2},{3},{1,2}}
  {{2},{1,3},{2,3}}    {{2},{3},{1,3}}
  {{3},{1,2},{1,3}}    {{1},{2},{1,2,3}}
  {{3},{1,2},{2,3}}    {{1},{3},{1,2,3}}
  {{3},{1,3},{2,3}}    {{2},{3},{1,2,3}}
  {{1},{1,2},{1,2,3}}
  {{1},{1,3},{1,2,3}}
  {{2},{1,2},{1,2,3}}
  {{2},{2,3},{1,2,3}}
  {{3},{1,3},{1,2,3}}
  {{3},{2,3},{1,2,3}}

			Triangle begins:
  1;
  0,     1;
  0,     2,     1;
  0,    15,     9,    1;
  0,   316,   198,   28,  1;
  0, 16885, 10710, 1610, 75, 1;
  ...

E. de Panafieu and S. Dovgal, Symbolic method and directed graph enumeration, arXiv:1903.09454 [math.CO], 2019.
R. W. Robinson, Counting digraphs with restrictions on the strong components, Combinatorics and Graph Theory '95 (T.-H. Ku, ed.), World Scientific, Singapore (1995), 343-354.

Cf. A058876 (mirror), A361579, A224069.
Row-sums are A003024, unlabeled A003087.
Column k = 1 is A003025(n) = |n*A134531(n)|.
Column k = n-1 is A058877.
For fixed sinks we get A368602.
A058891 counts set-systems, unlabeled A000612.
A323818 counts covering connected set-systems, unlabeled A323819.
Cf. A000169, A059201, A082402, A088957, A133686, A334282, A350415, A367904, A367908, A368600, A368601.

nn = 8; B[n_] := n! 2^Binomial[n, 2] ;ggf[egf_] := Normal[Series[egf, {z, 0, nn}]] /. Table[z^i -> z^i/2^Binomial[i, 2], {i, 0, nn}];Table[Take[(Table[B[n], {n, 0, nn}] CoefficientList[ Series[ggf[Exp[(u - 1) z]]/ggf[Exp[-z]], {z, 0, nn}], {z, u}])[[i]], i], {i, 1, nn + 1}] // Grid
nv=4;Table[Length[Select[Subsets[Subsets[Range[n]],{n}], Count[#,{_}]==k&&Length[Select[Tuples[#], UnsameQ@@#&]]==1&]],{n,0,nv},{k,0,n}]

T(n,k) = A368602(n,k) * binomial(n,k). - Gus Wiseman, Jan 03 2024

A368602 Triangle read by rows where T(n,k) is the number of labeled acyclic digraphs on {1..n} with sinks {1..k}.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 5, 3, 1, 0, 79, 33, 7, 1, 0, 3377, 1071, 161, 15, 1, 0, 362431, 92289, 10591, 705, 31, 1, 0, 93473345, 19856703, 1832705, 93375, 2945, 63, 1, 0, 56272471039, 10249747713, 789619327, 32382465, 782719, 12033, 127, 1

Offset: 0

Gus Wiseman, Jan 02 2024

Also the number of set-systems with n vertices and n edges such that {i} is a singleton edge iff i <= k, and such that there is only one way to choose a different vertex from each edge.

			Triangle begins:
    1
    0    1
    0    1    1
    0    5    3    1
    0   79   33    7    1
    0 3377 1071  161   15    1
    ...
Row n = 3 counts the following set-systems:
  {{1},{1,2},{1,3}}    {{1},{2},{1,3}}    {{1},{2},{3}}
  {{1},{1,2},{2,3}}    {{1},{2},{2,3}}
  {{1},{1,3},{2,3}}    {{1},{2},{1,2,3}}
  {{1},{1,2},{1,2,3}}
  {{1},{1,3},{1,2,3}}

Column k = n-1 is A000225 = A058877(n)/n.
Column k = 1 is A134531 (up to sign) or A003025(n)/n, non-fixed A350415.
For any choice of k sinks we get A361718.
A058891 counts set-systems, unlabeled A000612.
A059201 counts covering T_0 set-systems.
A323818 counts covering connected set-systems, unlabeled A323819.
Cf. A000169, A003024, A003087, A082402, A088957, A334282, A367862, A367904, A367908, A368600, A368601.

Table[Length[Select[Subsets[Subsets[Range[n]],{n}], Union@@Cases[#,{_}]==Range[k] && Length[Select[Tuples[#],UnsameQ@@#&]]==1&]], {n,0,3},{k,0,n}]

T(n,k) = A361718(n,k)/binomial(n,k).

More terms from Alois P. Heinz, Jan 04 2024

A370818 Number of sets of nonempty subsets of {1..n} with only one possible way to choose a set of different vertices of each edge.

Original entry on oeis.org

1, 2, 6, 45, 1352, 157647, 63380093, 85147722812, 385321270991130

Offset: 0

Gus Wiseman, Mar 12 2024

			The set-system {{2},{1,2},{2,4},{1,3,4}} has unique choice (2,1,4,3) so is counted under a(4).

This is the unique version of A367902, complement A367903.
Choosing a sequence gives A367904, ranks A367908.
The maximal case is A368601, complement A368600.
This is the restriction of A370638 to A000225.
Factorizations of this type are counted by A370645.
A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.
A058891 counts set-systems, A003465 covering, A323818 connected.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
Cf. A133686, A134964, A367772, A367867, A368101, A368109, A370584.

Table[Length[Select[Subsets[Subsets[Range[n]]], Length[Union[Sort/@Select[Tuples[#],UnsameQ@@#&]]]==1&]],{n,0,3}]

a(n) = A370638(2^n - 1).

Binomial transform of A368601. - Christian Sievers, Aug 12 2024

a(5)-a(8) from Christian Sievers, Aug 12 2024

cp's OEIS Frontend

A369143 Number of labeled simple graphs with n edges and n vertices such that it is not possible to choose a different vertex from each edge (non-choosable).

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A369144 Number of labeled simple graphs with n edges covering n vertices such that it is not possible to choose a different vertex from each edge (non-choosable).

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A369201 Number of unlabeled simple graphs with n vertices and n edges such that it is not possible to choose a different vertex from each edge (non-choosable).

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A361718 Triangular array read by rows. T(n,k) is the number of labeled directed acyclic graphs on [n] with exactly k nodes of indegree 0.

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A368602 Triangle read by rows where T(n,k) is the number of labeled acyclic digraphs on {1..n} with sinks {1..k}.

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A370818 Number of sets of nonempty subsets of {1..n} with only one possible way to choose a set of different vertices of each edge.

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