A324326 -id:A324326 - 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

A324324 MM-numbers of crossing set partitions.

Original entry on oeis.org

2117, 3973, 4843, 5891, 6757, 7181, 7801, 10019, 10063, 11051, 11567, 13021, 13193, 13459, 14123, 14921, 17603, 18407, 18761, 18877, 19307, 19633, 20941, 21083, 21251, 21457, 22849, 23519, 23533, 24727, 26101, 27133, 27169, 27173, 27413, 29111, 30479, 31261

Offset: 1

Gus Wiseman, Feb 22 2019

nonn

A multiset multisystem is a finite multiset of finite multisets. 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 formed by taking the multiset of prime indices of each part in the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset multisystem with MM-number 78 is {{},{1},{1,2}}.

A multiset multisystem is crossing if it contains two parts of the form {{...x...y...},{...z...t...}} with x < z < y < t or z < x < t < y.

Cf. A000108 (non-crossing set partitions), A001055, A001222, A003963, A005117, A016098 (crossing set partitions), A054726, A056239, A112798, A302242, A302243, A302505, A302521 (MM-numbers of set partitions).

Cf. A324166, A324167, A324169, A324170, A324171, A324326.

croXQ[stn_]:=MatchQ[stn,{_,{_,x_,_,y_,_},_,{_,z_,_,t_,_},_}/;xTable[PrimePi[p],{k}]]]];
setptnQ[bks_]:=UnsameQ@@Join@@bks&&!MemberQ[bks,{}];
Select[Range[10000],And[croXQ[primeMS/@primeMS[#]],setptnQ[primeMS/@primeMS[#]]]&]

A324325 Number of non-crossing multiset partitions of a multiset whose multiplicities are the prime indices of n.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 5, 5, 9, 7, 7, 11, 11, 12, 16, 14, 15, 26, 22, 21, 29, 19, 30, 33, 31, 30, 66, 38, 42, 52, 56, 42, 47, 45, 57, 82, 77, 67, 77, 67, 101, 98, 135, 64, 137, 97, 176, 104, 109, 109, 118, 105, 231, 213, 97, 127, 181, 139, 297, 173, 385, 195, 269

Offset: 1

Gus Wiseman, Feb 22 2019

nonn

This multiset (row n of A305936) is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

A multiset partition is crossing if it contains two blocks of the form {{...x...y...},{...z...t...}} where x < z < y < t or z < x < t < y.

			The a(16) = 14 non-crossing multiset partitions of the multiset {1,2,3,4}:
  {{1,2,3,4}}
  {{1},{2,3,4}}
  {{2},{1,3,4}}
  {{3},{1,2,4}}
  {{4},{1,2,3}}
  {{1,2},{3,4}}
  {{1,4},{2,3}}
  {{1},{2},{3,4}}
  {{1},{3},{2,4}}
  {{1},{4},{2,3}}
  {{2},{3},{1,4}}
  {{2},{4},{1,3}}
  {{3},{1,2},{4}}
  {{1},{2},{3},{4}}
Missing from this list is {{1,3},{2,4}}.

Cf. A000108, A001055, A001970, A016098, A054726, A099947, A181821, A305936, A306438, A318284, A318285.

Cf. A324167, A324168, A324169, A324170, A324171, A324324, A324326.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
nonXQ[stn_]:=!MatchQ[stn,{_,{_,x_,_,y_,_},_,{_,z_,_,t_,_},_}/;x

a(n) + A324326(n) = A318284(n).

A326292 Number of crossing integer partitions of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 43, 57, 80, 105, 142, 186, 248, 320, 421, 539, 698, 889, 1140, 1438, 1827, 2291, 2882, 3593, 4489, 5559, 6902, 8503, 10484, 12853, 15763

Offset: 0

Gus Wiseman, Oct 03 2019

nonn

A multiset partition is crossing if it has two blocks of the form {...x...y...}, {...z...t...} where x < z < y < t or z < x < t < y. An integer partition is crossing if, by replacing each part with its multiset of prime indices, we obtain a crossing multiset partition.

			The a(31) = 1 through a(36) = 7 partitions:
  21,10  21,10,1  21,10,2    21,10,3      21,10,4        21,10,5
                  21,10,1,1  21,10,2,1    21,10,2,2      21,10,3,2
                             21,10,1,1,1  21,10,3,1      21,10,4,1
                                          21,10,2,1,1    21,10,2,2,1
                                          21,10,1,1,1,1  21,10,3,1,1
                                                         21,10,2,1,1,1
                                                         21,10,1,1,1,1,1

The Heinz numbers of these partitions are given by A324170.

Cf. A000108, A002662, A016098, A056239, A112798, A324172, A324326, A326210, A326252, A326332.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
croXQ[stn_]:=MatchQ[stn,{_,{_,x_,_,y_,_},_,{_,z_,_,t_,_},_}/;x

More terms from Jinyuan Wang, Jun 28 2020

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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A324324 MM-numbers of crossing set partitions.

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A324325 Number of non-crossing multiset partitions of a multiset whose multiplicities are the prime indices of n.

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A326292 Number of crossing integer partitions of n.

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