A016098 -id:A016098 - cp's OEIS frontend

A326210 Number of labeled simple graphs with vertices {1..n} containing a nesting pair of edges, where two edges {a,b}, {c,d} are nesting if a < c and b > d or a > c and b < d.

0, 0, 0, 0, 16, 672, 29888, 2071936, 268204288, 68717285888, 35184350796800, 36028796807919616, 73786976292712960000, 302231454903635611721728, 2475880078570760326175178752, 40564819207303340845566684397568, 1329227995784915872903782635437883392

Offset: 0

Gus Wiseman, Jun 19 2019

nonn

Also simple graphs containing a crossing pair of edges, where two edges {a,b}, {c,d} are crossing if a < c < b < d or c < a < d < b.

Also simple graphs such that, if the edges are listed in lexicographic order, their maxima (seconds) are not weakly increasing.

			The a(4) = 16 nesting edge-sets:
  {14,23}
  {12,14,23}
  {13,14,23}
  {14,23,24}
  {14,23,34}
  {12,13,14,23}
  {12,14,23,24}
  {12,14,23,34}
  {13,14,23,24}
  {13,14,23,34}
  {14,23,24,34}
  {12,13,14,23,24}
  {12,13,14,23,34}
  {12,14,23,24,34}
  {13,14,23,24,34}
  {12,13,14,23,24,34}
The a(4) = 16 crossing edge-sets:
  {13,24}
  {12,13,24}
  {13,14,24}
  {13,23,24}
  {13,24,34}
  {12,13,14,24}
  {12,13,23,24}
  {12,13,24,34}
  {13,14,23,24}
  {13,14,24,34}
  {13,23,24,34}
  {12,13,14,23,24}
  {12,13,14,24,34}
  {12,13,23,24,34}
  {13,14,23,24,34}
  {12,13,14,23,24,34}

Andrew Howroyd, Table of n, a(n) for n = 0..50

Non-nesting graphs are A054726.
Nesting digraphs are A326209.
Nesting (or crossing) set partitions are A016098.
MM-numbers of nesting multiset partitions are A326256.
Cf. A001519, A006125, A095661, A324170, A324172, A324326.
Cf. A326211, A326243, A326244, A326248, A326250.

Table[Length[Select[Subsets[Subsets[Range[n],{2}]],!OrderedQ[Last/@#]&]],{n,0,5}]

seq(n)={my(p=1 + 3/2*x - x^2 - x/2*sqrt(1 - 12*x + 4*x^2 + O(x^n))); concat([0], vector(n, k, 2^binomial(k,2)-polcoef(p,k)))} \\ Andrew Howroyd, Aug 26 2019

A006125(n) = a(n) + A054726(n).

Terms a(7) and beyond from Andrew Howroyd, Aug 26 2019

A326243 Number of capturing set partitions of {1..n}.

Original entry on oeis.org

0, 0, 0, 0, 1, 11, 80, 503, 2993, 17609, 105017, 644528, 4107600, 27313805, 189866541, 1379728831, 10470032837, 82833202559, 681977545967, 5832430910181, 51723181525978, 474866750479993, 4506706112772881, 44151975623559477, 445958774322599940, 4638590033810841345

Offset: 0

Gus Wiseman, Jun 19 2019

nonn

A set partition is capturing if it has two blocks of the form {...x...y...}, {...z...t...} where x < z < t < y or z < x < y < t. This is a weaker condition than nesting, so for example {{1,3,5},{2,4}} is capturing but not nesting.

			The a(5) = 11 capturing set partitions:
  {{1,2,5},{3,4}}
  {{1,3,4},{2,5}}
  {{1,3,5},{2,4}}
  {{1,4},{2,3,5}}
  {{1,4,5},{2,3}}
  {{1,5},{2,3,4}}
  {{1},{2,5},{3,4}}
  {{1,4},{2,3},{5}}
  {{1,5},{2},{3,4}}
  {{1,5},{2,3},{4}}
  {{1,5},{2,4},{3}}

Eric Marberg, Crossings and nestings in colored set partitions, arXiv preprint arXiv:1203.5738 [math.CO], 2012.

Non-capturing set partitions are A054391.
Crossing and nesting set partitions are (both) A016098.
Cf. A000108, A000110, A001519, A054391, A058681, A122880, A324170.
Cf. A326209, A326211, A326237, A326246, A326249, A326255, A326259.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
capXQ[stn_]:=MatchQ[stn,{_,{_,x_,_,y_,_},_,{_,z_,_,t_,_},_}/;xt||x>z&&y

a(n) = A000110(n) - A054391(n).

a(12) and beyond from Christian Sievers, Aug 23 2024

A136653 G.f.: A(x) satisfies: coefficient of x^n in A(x)^(n+1)/(n+1) = 2^(n*(n-1)/2).

Original entry on oeis.org

1, 1, 1, 4, 39, 748, 27162, 1880872, 252273611, 66358216668, 34506398937158, 35644762692112792, 73356520492898454022, 301274559225693420690360, 2471654510727312089903896948, 40527708183358718551543295827536, 1328579216048284168977214446788083699

Offset: 0

Paul D. Hanna, Jan 15 2008

nonn

a(n) is the number of graphs on vertices 1,...,n such that, when these vertices are arranged counterclockwise around a circle and edges are drawn as straight line segments, the resulting diagram is connected. - Jonathan Novak (j2novak(AT)math.uwaterloo.ca), Apr 30 2010

In this interpretation, both intersecting (set theoretically) and crossing (topologically) edges are considered connected. - Gus Wiseman, Feb 23 2019

			G.f.: A(x) = 1 + x + x^2 + 4*x^3 + 39*x^4 + 748*x^5 + 27162*x^6 +...
Let F(x) = 1 + x + 2*x^2 + 8*x^3 + 64*x^4 + 1024*x^5 +...+ 2^(n*(n-1)/2)*x^n +..
then A(x) = F(x/A(x)), A(x*F(x)) = F(x).
Coefficient of x^n in A(x)^(n+1)/(n+1) = 2^(n*(n-1)/2),
as can be seen by the main diagonal in the array of
coefficients in the initial powers of A(x):
A^1: [(1), 1, 1, 4, 39, 748, 27162, 1880872, 252273611,...;
A^2: [1, (2), 3, 10, 87, 1582, 55914, 3817876, 508370795,...;
A^3: [1, 3, (6), 19, 147, 2517, 86398, 5813550, 768378627,...;
A^4: [1, 4, 10, (32), 223, 3572, 118778, 7870640, 1032387787,...;
A^5: [1, 5, 15, 50, (320), 4771, 153245, 9992130, 1300492845,...;
A^6: [1, 6, 21, 74, 444, (6144), 190023, 12181278, 1572792585,...;
A^7: [1, 7, 28, 105, 602, 7728, (229376), 14441659, 1849390375,...;
A^8: [1, 8, 36, 144, 802, 9568, 271616, (16777216), 2130394591,...;
A^9: [1, 9, 45, 192, 1053, 11718, 317112, 19192320, (2415919104),...;
dividing each diagonal term in row n by (n+1) gives 2^(n*(n-1)/2).
The diagonal above the main diagonal gives coefficients of l.g.f.:
log(F(x)) = x + 3*x^2/2 + 19*x^3/3 + 223*x^4/4 + 4771*x^5/5 +...

Andrew Howroyd, Table of n, a(n) for n = 0..50
Gus Wiseman, The a(4) = 39 graphs connected by overlaps or crossings.

Cf. A136652 (log(A(x))), A136654.
Cf. A000699, A002662, A006125, A007297, A016098, A054726, A099947, A306438.
Cf. A324166, A324169, A324172, A324173, A324327, A324328.

max = 15; s = x*Sum[2^(k*(k-1)/2)*x^k, {k, 0, max}] + O[x]^(max+2); x/InverseSeries[s] + O[x]^(max+1) // CoefficientList[#, x]& (* Jean-François Alcover, Sep 03 2017 *)
croXQ[stn_]:=MatchQ[stn,{_,{_,x_,_,y_,_},_,{_,z_,_,t_,_},_}/;x0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
bicmpts[stn_]:=csm[Union[Subsets[stn,{1}],Select[Subsets[stn,{2}],Intersection@@#!={}&],Select[Subsets[stn,{2}],croXQ]]];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],And[Union@@#==Range[n],Length[bicmpts[#]]<=1]&]],{n,0,5}] (* Gus Wiseman, Feb 23 2019 *)

a(n)=polcoeff(x/serreverse(x*sum(k=0,n,2^(k*(k-1)/2)*x^k +x*O(x^n))),n)

G.f.: A(x) = x/Series_Reversion( x*Sum_{k=0..n} 2^(k(k-1)/2)*x^k ).

Equals the free cumulant sequence corresponding to A006125. - Jonathan Novak (j2novak(AT)math.uwaterloo.ca), Apr 30 2010

Name changed and part of prior name moved to formula section by Paul D. Hanna, Sep 19 2013

A324172 Number of subsets of {1,...,n} that cross their complement.

Original entry on oeis.org

0, 0, 0, 0, 2, 10, 32, 84, 198, 438, 932, 1936, 3962, 8034, 16200, 32556, 65294, 130798, 261836, 523944, 1048194, 2096730, 4193840, 8388100, 16776662, 33553830, 67108212, 134217024, 268434698, 536870098, 1073740952, 2147482716, 4294966302, 8589933534, 17179868060

Offset: 0

Gus Wiseman, Feb 17 2019

Two sets cross each other if they are of the form {{...x...y...}, {...z...t...}} where x < z < y < t or z < x < t < y.

Also the number of verex cuts in the wheel graph on n nodes. - Eric W. Weisstein, Apr 22 2023

			The a(5) = 10 subsets are {1,3}, {1,4}, {2,4}, {2,5}, {3,5}, {1,2,4}, {1,3,4}, {1,3,5}, {2,3,5}, {2,4,5}.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-9,7,-2).
Eric Weisstein's World of Mathematics, Vertex Cut
Eric Weisstein's World of Mathematics, Wheel Graph

Cf. A000108, A000110, A000124, A001263, A002061, A002662, A016098, A306438.

Cf. A324166, A324167, A324168, A324169, A324173.

croXQ[stn_]:=MatchQ[stn,{_,{_,x_,_,y_,_},_,{_,z_,_,t_,_},_}/;x

concat([0,0,0,0], Vec(2*x^4 / ((1 - x)^3*(1 - 2*x)) + O(x^40))) \\ Colin Barker, Feb 19 2019

a(0) = 0; a(n) = 2^n - n^2 + n - 2.
a(n) = 2*A002662(n-1) for n > 0.
G.f.: 2*x^4/((1-2*x)*(1-x)^3).
a(n) = 5*a(n-1) - 9*a(n-2) + 7*a(n-3) - 2*a(n-4) for n>4. - Colin Barker, Feb 18 2019

A326211 Number of unsortable normal multiset partitions of weight n.

Original entry on oeis.org

0, 0, 0, 1, 17, 170, 1455, 11678, 92871, 752473

Offset: 0

Gus Wiseman, Jun 19 2019

A multiset partition is normal if it covers an initial interval of positive integers. It is unsortable if no permutation has an ordered concatenation, or equivalently if the concatenation of its lexicographically-ordered parts is not weakly increasing. For example, the multiset partition {{1,2},{1,1,1},{2,2,2}} is sortable because the permutation ((1,1,1),(1,2),(2,2,2)) has concatenation (1,1,1,1,2,2,2,2), which is weakly increasing.

			The a(3) = 1 and a(4) = 17 multiset partitions:
  {{1,3},{2}}  {{1,1,3},{2}}
               {{1,2},{1,2}}
               {{1,2},{1,3}}
               {{1,2,3},{2}}
               {{1,2,4},{3}}
               {{1,3},{2,2}}
               {{1,3},{2,3}}
               {{1,3},{2,4}}
               {{1,3,3},{2}}
               {{1,3,4},{2}}
               {{1,4},{2,3}}
               {{1},{1,3},{2}}
               {{1},{2,4},{3}}
               {{1,3},{2},{2}}
               {{1,3},{2},{3}}
               {{1,3},{2},{4}}
               {{1,4},{2},{3}}

Unsortable set partitions are A058681.
Sortable normal multiset partitions are A326212.
Non-crossing normal multiset partitions are A324171.
MM-numbers of unsortable multiset partitions are A326258.
Cf. A000108, A016098, A255906, A324170.
Cf. A326209, A326210, A326243, A326250, A326255, A326256.

lexsort[f_,c_]:=OrderedQ[PadRight[{f,c}]];
allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
Table[Length[Select[Sort[#,lexsort]&/@Join@@mps/@allnorm[n],!OrderedQ[Join@@#]&]],{n,0,5}]

A255906(n) = a(n) + A326212(n).

A326244 Number of labeled n-vertex simple graphs without crossing or nesting edges.

Original entry on oeis.org

1, 1, 2, 8, 36, 160, 704, 3088, 13536, 59328, 260032, 1139712

Offset: 0

Gus Wiseman, Jun 20 2019

Two edges {a,b}, {c,d} are crossing if a < c < b < d or c < a < d < b, and nesting if a < c < d < b or c < a < b < d.

Eric Marberg, Crossings and nestings in colored set partitions, arXiv preprint arXiv:1203.5738 [math.CO], 2012.
Gus Wiseman, The a(4) = 36 non-crossing, non-nesting simple labeled graphs.

The case for set partitions is A001519.
Simple graphs with crossing or nesting edges are A326279.
Cf. A000088, A000108, A006125, A016098, A054726, A095661, A117662.
Cf. A326209, A326210, A326250, A326255, A326256.

croXQ[stn_]:=MatchQ[stn,{_,{x_,y_},_,{z_,t_},_}/;x_,{x_,y_},_,{z_,t_},_}/;x

A006125(n) = a(n) + A326279(n).
Conjectures from Colin Barker, Jun 28 2019: (Start)
G.f.: (1 - x)*(1 - 4*x) / (1 - 6*x + 8*x^2 - 4*x^3).
a(n) = 6*a(n-1) - 8*a(n-2) + 4*a(n-3) for n>2.
(End)

A326256 MM-numbers of nesting multiset partitions.

Original entry on oeis.org

667, 989, 1334, 1633, 1769, 1817, 1978, 2001, 2021, 2323, 2461, 2623, 2668, 2967, 2987, 3197, 3266, 3335, 3538, 3634, 3713, 3749, 3956, 3979, 4002, 4042, 4171, 4331, 4379, 4429, 4439, 4577, 4646, 4669, 4747, 4819, 4859, 4899, 4922, 4945, 5029, 5246, 5267, 5307

Offset: 1

Gus Wiseman, Jun 20 2019

nonn

First differs from A326255 in lacking 2599.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The multiset multisystem with MM-number n is obtained by taking the multiset of prime indices of each prime index of n.
A multiset partition is nesting if it has two blocks of the form {...x,y...}, {...z,t...} where x < z and t < y or z < x and y < t. This is a stronger condition than capturing, so for example {{1,3,5},{2,4}} is capturing but not nesting.

			The sequence of terms together with their multiset multisystems begins:
   667: {{2,2},{1,3}}
   989: {{2,2},{1,4}}
  1334: {{},{2,2},{1,3}}
  1633: {{2,2},{1,1,3}}
  1769: {{1,3},{1,2,2}}
  1817: {{2,2},{1,5}}
  1978: {{},{2,2},{1,4}}
  2001: {{1},{2,2},{1,3}}
  2021: {{1,4},{2,3}}
  2323: {{2,2},{1,6}}
  2461: {{2,2},{1,1,4}}
  2623: {{1,4},{1,2,2}}
  2668: {{},{},{2,2},{1,3}}
  2967: {{1},{2,2},{1,4}}
  2987: {{1,3},{2,2,2}}
  3197: {{2,2},{1,7}}
  3266: {{},{2,2},{1,1,3}}
  3335: {{2},{2,2},{1,3}}
  3538: {{},{1,3},{1,2,2}}
  3634: {{},{2,2},{1,5}}

MM-numbers of crossing multiset partitions are A324170.
MM-numbers of capturing multiset partitions are A326255.
Nesting set partitions are A016098.
Capturing set partitions are A326243.
Cf. A001055, A034827, A058681, A112798, A117662, A302242.
Cf. A326211, A326248, A326257, A326258, A326260.

nesXQ[stn_]:=MatchQ[stn,{_,{_,x_,y_,_},_,{_,z_,t_,_},_}/;(xt)||(x>z&&yTable[PrimePi[p],{k}]]]];
Select[Range[10000],nesXQ[primeMS/@primeMS[#]]&]

A326209 Number of nesting labeled digraphs with vertices {1..n}.

Original entry on oeis.org

0, 0, 4, 408, 64528

Offset: 0

Gus Wiseman, Jun 19 2019

Two edges (a,b), (c,d) are nesting if a < c and b > d or a > c and b < d.
Also unsortable digraphs with vertices {1..n}, where a digraph is sortable if, when the edges are listed in lexicographic order, their targets are weakly increasing.
Also the number of semicrossing digraphs with vertices {1..n}, where two edges (a,b), (c,d) are semicrossing if a < c and b < d or a > c and b > d. For example, the a(2) = 4 semicrossing digraph edge-sets are:
  {11,22}
  {11,12,22}
  {11,21,22}
  {11,12,21,22}

			The a(2) = 4 nesting digraph edge-sets:
  {12,21}
  {11,12,21}
  {12,21,22}
  {11,12,21,22}

Non-nesting digraphs are A326237.
Nesting set partitions are A016098.
MM-numbers of nesting multiset partitions are A326256.
MM-numbers of unsortable multiset partitions are A326258.
Cf. A000108, A001519, A002416, A229865.
Cf. A326210, A326211, A326243, A326246, A326248, A326250.

Table[Length[Select[Subsets[Tuples[Range[n],2]],!OrderedQ[Last/@#]&]],{n,4}]

A002416(n) = a(n) + A326237(n).

A326258 MM-numbers of unsortable multiset partitions (with empty parts allowed).

Original entry on oeis.org

145, 169, 215, 290, 338, 355, 377, 395, 430, 435, 473, 481, 505, 507, 535, 559, 565, 580, 645, 667, 676, 695, 710, 725, 754, 790, 793, 803, 815, 841, 845, 860, 865, 869, 870, 905, 923, 946, 962, 965, 989, 995, 1010, 1014, 1015, 1027, 1065, 1070, 1073, 1075

Offset: 1

Gus Wiseman, Jun 22 2019

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The multiset multisystem with MM-number n is obtained by taking the multiset of prime indices of each prime index of n.

A multiset partition is unsortable if no permutation has an ordered concatenation. For example, the multiset partition ((1,2),(1,1,1),(2,2,2)) is sortable because the permutation ((1,1,1),(1,2),(2,2,2)) has concatenation (1,1,1,1,2,2,2,2), which is weakly increasing.

			The sequence of terms together with their multiset multisystems begins:
  145: {{2},{1,3}}
  169: {{1,2},{1,2}}
  215: {{2},{1,4}}
  290: {{},{2},{1,3}}
  338: {{},{1,2},{1,2}}
  355: {{2},{1,1,3}}
  377: {{1,2},{1,3}}
  395: {{2},{1,5}}
  430: {{},{2},{1,4}}
  435: {{1},{2},{1,3}}
  473: {{3},{1,4}}
  481: {{1,2},{1,1,2}}
  505: {{2},{1,6}}
  507: {{1},{1,2},{1,2}}
  535: {{2},{1,1,4}}
  559: {{1,2},{1,4}}
  565: {{2},{1,2,3}}
  580: {{},{},{2},{1,3}}
  645: {{1},{2},{1,4}}
  667: {{2,2},{1,3}}

Unsortable set partitions are A058681.
Normal unsortable multiset partitions are A326211.
Unsortable digraphs are A326209.
MM-numbers of crossing multiset partitions are A324170.
MM-numbers of nesting multiset partitions are A326256.
MM-numbers of capturing multiset partitions are A326255.
Cf. A001055, A016098, A056239, A112798, A302242.
Cf. A326212, A326243, A326257, A326259.

lexsort[f_,c_]:=OrderedQ[PadRight[{f,c}]];
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],!OrderedQ[Join@@Sort[primeMS/@primeMS[#],lexsort]]&]

A324166 Number of totally crossing set partitions of {1,...,n}.

Original entry on oeis.org

1, 1, 1, 1, 2, 6, 18, 57, 207, 842, 3673, 17062, 84897

Offset: 0

Gus Wiseman, Feb 17 2019

A set partition is totally crossing if every pair of distinct blocks is of the form {{...x...y...}, {...z...t...}} for some x < z < y < t or z < x < t < y.

			The a(6) = 18 totally crossing set partitions:
  {{1,2,3,4,5,6}}
  {{1,4,6},{2,3,5}}
  {{1,4,5},{2,3,6}}
  {{1,3,6},{2,4,5}}
  {{1,3,5},{2,4,6}}
  {{1,3,4},{2,5,6}}
  {{1,2,5},{3,4,6}}
  {{1,2,4},{3,5,6}}
  {{4,6},{1,2,3,5}}
  {{3,6},{1,2,4,5}}
  {{3,5},{1,2,4,6}}
  {{2,6},{1,3,4,5}}
  {{2,5},{1,3,4,6}}
  {{2,4},{1,3,5,6}}
  {{1,5},{2,3,4,6}}
  {{1,4},{2,3,5,6}}
  {{1,3},{2,4,5,6}}
  {{1,4},{2,5},{3,6}}

Cf. A000108 (non-crossing partitions), A000110, A000296, A002662, A016098 (crossing partitions), A054726, A099947 (topologically connected partitions), A305854, A306006, A306418, A306438, A319752.

Cf. A324170, A324171, A324172, A324173.

nn=6;
nonXQ[stn_]:=!MatchQ[stn,{_,{_,x_,_,y_,_},_,{_,z_,_,t_,_},_}/;x

cp's OEIS Frontend

A326210 Number of labeled simple graphs with vertices {1..n} containing a nesting pair of edges, where two edges {a,b}, {c,d} are nesting if a < c and b > d or a > c and b < d.

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A326243 Number of capturing set partitions of {1..n}.

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A136653 G.f.: A(x) satisfies: coefficient of x^n in A(x)^(n+1)/(n+1) = 2^(n*(n-1)/2).

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A324172 Number of subsets of {1,...,n} that cross their complement.

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A326211 Number of unsortable normal multiset partitions of weight n.

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A326244 Number of labeled n-vertex simple graphs without crossing or nesting edges.

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A326256 MM-numbers of nesting multiset partitions.

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Mathematica