A326256 -id:A326256 - cp's OEIS frontend

A326250 Number of weakly nesting simple graphs with vertices {1..n}.

0, 0, 0, 3, 50, 982, 32636, 2096723

Offset: 0

Gus Wiseman, Jun 21 2019

Two edges {a,b}, {c,d} are weakly nesting if a <= c < d <= b or c <= a < b <= d.

Gus Wiseman, The a(4) = 50 weakly nesting simple graphs.

Non-nesting set partitions are A000108.
Non-crossing graphs are A054726.
Nesting digraphs are A326209.
Crossing graphs are A326210.
MM-numbers of nesting multiset partitions are A326256.
Cf. A000088, A000332, A006125, A117662.
Cf. A326211, A326243, A326244, A326248, A326257, A326279.

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

Conjecture: A006125(n) = a(n) + A000108(n).

A326249 Number of capturing set partitions of {1..n} that are not nesting.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 9, 55, 283, 1324, 5838, 24744

Offset: 0

Gus Wiseman, Jun 20 2019

Capturing is a weaker condition than nesting. 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, and nesting if it has two blocks of the form {...x,y...}, {...z,t...} where x < z < t < y or z < x < y < t. For example, {{1,3,5},{2,4}} is capturing but not nesting, so is counted under a(5).

			The a(6) = 9 set partitions:
  {{1},{2,4,6},{3,5}}
  {{1,3,5},{2,4},{6}}
  {{1,3,6},{2,4},{5}}
  {{1,3,6},{2,5},{4}}
  {{1,4,6},{2},{3,5}}
  {{1,4,6},{2,5},{3}}
  {{1,3,5},{2,4,6}}
  {{1,2,4,6},{3,5}}
  {{1,3,5,6},{2,4}}

MM-numbers of capturing, non-nesting multiset partitions are A326260.
Nesting set partitions are A016098.
Capturing set partitions are A326243.
Non-crossing, nesting set partitions are A122880 (conjectured).
Cf. A000110, A054391, A058681, A095661, A099947.
Cf. A326212, A326237, A326245, A326246, A054391, A326255, A326256.

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_,_},_}/;x_,{_,x_,y_,_},_,{_,z_,t_,_},_}/;x

A326259 MM-numbers of crossing, capturing multiset partitions (with empty parts allowed).

Original entry on oeis.org

8903, 15167, 16717, 17806, 18647, 20329, 20453, 21797, 22489, 25607, 26709, 27649, 29551, 30334, 31373, 32741, 33434, 34691, 35177, 35612, 35821, 37091, 37133, 37294, 37969, 38243, 39493, 40658, 40906, 41449, 42011, 42949, 43594, 43817, 43873, 44515, 44861

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

			The sequence of terms together with their multiset multisystems begins:
   8903: {{1,3},{2,2,4}}
  15167: {{1,3},{2,2,5}}
  16717: {{2,4},{1,3,3}}
  17806: {{},{1,3},{2,2,4}}
  18647: {{1,3},{2,2,6}}
  20329: {{1,3},{1,2,2,4}}
  20453: {{1,2,3},{1,2,4}}
  21797: {{1,1,3},{2,2,4}}
  22489: {{1,4},{2,2,5}}
  25607: {{1,3},{2,2,7}}
  26709: {{1},{1,3},{2,2,4}}
  27649: {{1,4},{2,2,6}}
  29551: {{1,3},{2,2,8}}
  30334: {{},{1,3},{2,2,5}}
  31373: {{2,5},{1,3,3}}
  32741: {{1,3},{2,2,2,4}}
  33434: {{},{2,4},{1,3,3}}
  34691: {{1,2,3},{2,2,4}}
  35177: {{1,3},{1,2,2,5}}
  35612: {{},{},{1,3},{2,2,4}}

Crossing set partitions are A000108.
Capturing set partitions are A326243.
Crossing, capturing set partitions are A326246.
MM-numbers of crossing multiset partitions are A324170.
MM-numbers of nesting multiset partitions are A326256.
MM-numbers of capturing multiset partitions are A326255.
MM-numbers of unsortable multiset partitions are A326258.
Cf. A001055, A001519, A016098, A056239, A058681, A112798, A122880, A302242.
Cf. A326211, A326245, A326248, A326249, A054391.

croXQ[stn_]:=MatchQ[stn,{_,{_,x_,_,y_,_},_,{_,z_,_,t_,_},_}/;x_,{_,x_,_,y_,_},_,{_,z_,_,t_,_},_}/;xTable[PrimePi[p],{k}]]]];
Select[Range[100000],capXQ[primeMS/@primeMS[#]]&&croXQ[primeMS/@primeMS[#]]&]

A326279 Number of labeled n-vertex simple graphs containing either a crossing or a nesting pair of edges.

Original entry on oeis.org

0, 0, 0, 0, 28, 864, 32064, 2094064

Offset: 0

Gus Wiseman, Jun 23 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.

			The a(4) = 28 edge-sets:
  {13,24}  {12,13,24}  {12,13,14,23}  {12,13,14,23,24}  {12,13,14,23,24,34}
  {14,23}  {12,14,23}  {12,13,14,24}  {12,13,14,23,34}
           {13,14,23}  {12,13,23,24}  {12,13,14,24,34}
           {13,14,24}  {12,13,24,34}  {12,13,23,24,34}
           {13,23,24}  {12,14,23,24}  {12,14,23,24,34}
           {13,24,34}  {12,14,23,34}  {13,14,23,24,34}
           {14,23,24}  {13,14,23,24}
           {14,23,34}  {13,14,23,34}
                       {13,14,24,34}
                       {13,23,24,34}
                       {14,23,24,34}

Crossing and nesting simple graphs are (both) A326210, while non-crossing, non-nesting simple graphs are A326244.
Cf. A000108, A001519, A006125, A016098, A054726, A095661, A324170.
Cf. A326209, A326211, A326248, A326250, A326256.

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

A006125(n) = a(n) + A326244(n).

A326260 MM-numbers of capturing, non-nesting multiset partitions (with empty parts allowed).

Original entry on oeis.org

2599, 4163, 5198, 6463, 6893, 7291, 7797, 8326, 8507, 9131, 9959, 10396, 10649, 11041, 11639, 12489, 12811, 12926, 12995, 13786, 14237, 14582, 14899, 15157, 15594, 16123, 16403, 16652, 17014, 17063, 17089, 17141, 18101, 18193, 18262, 18643, 18659, 19337, 19389

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

			The sequence of terms together with their multiset multisystems begins:
   2599: {{2,2},{1,2,3}}
   4163: {{2,2},{1,2,4}}
   5198: {{},{2,2},{1,2,3}}
   6463: {{2,2},{1,1,2,3}}
   6893: {{1,2,2},{1,2,3}}
   7291: {{2,2},{1,2,5}}
   7797: {{1},{2,2},{1,2,3}}
   8326: {{},{2,2},{1,2,4}}
   8507: {{2,3},{1,2,4}}
   9131: {{2,2},{1,2,6}}
   9959: {{2,2},{1,1,2,4}}
  10396: {{},{},{2,2},{1,2,3}}
  10649: {{2,2},{1,2,2,3}}
  11041: {{1,2,2},{1,2,4}}
  11639: {{2,2,2},{1,2,3}}
  12489: {{1},{2,2},{1,2,4}}
  12811: {{2,2},{1,2,7}}
  12926: {{},{2,2},{1,1,2,3}}
  12995: {{2},{2,2},{1,2,3}}
  13786: {{},{1,2,2},{1,2,3}}

Non-nesting set partitions are A000108.
Capturing set partitions are A326243.
Capturing, non-nesting set partitions are A326249.
MM-numbers of nesting multiset partitions are A326256.
MM-numbers of capturing multiset partitions are A326255.
Cf. A001055, A034827, A056239, A058681, A112798, A117662, A302242, A324170.
Cf. A326212, A326246, A054391, A326257, A326258, A326259.

capXQ[stn_]:=MatchQ[stn,{_,{_,x_,_,y_,_},_,{_,z_,_,t_,_},_}/;x_,{_,x_,y_,_},_,{_,z_,t_,_},_}/;xTable[PrimePi[p],{k}]]]];
Select[Range[10000],!nesXQ[primeMS/@primeMS[#]]&&capXQ[primeMS/@primeMS[#]]&]

A326247 Number of labeled n-vertex 2-edge multigraphs that are neither crossing nor nesting.

Original entry on oeis.org

0, 0, 1, 9, 32, 80, 165, 301, 504, 792, 1185, 1705, 2376, 3224, 4277, 5565, 7120, 8976, 11169, 13737, 16720, 20160, 24101, 28589, 33672, 39400, 45825, 53001, 60984, 69832, 79605, 90365, 102176, 115104, 129217, 144585, 161280, 179376, 198949, 220077, 242840

Offset: 0

Gus Wiseman, Jun 20 2019

nonn

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.

			The a(3) = 9 pairs of edges:
  {12,12}
  {12,13}
  {12,23}
  {13,12}
  {13,13}
  {13,23}
  {23,12}
  {23,13}
  {23,23}

Gus Wiseman, The a(4) = 32 pairs of edges that are neither crossing nor nesting.

The case for simple graphs (rather than multigraphs) is A095661.
Simple graphs that are neither crossing nor nesting are A326244.
The case for set partitions is A001519.
Non-crossing and non-nesting simple graphs are (both) A054726.
Cf. A000108, A002061, A006125, A117662, A324170, A326250, A326256.

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

Conjectures from Colin Barker, Jun 21 2019: (Start)
G.f.: x^2*(1 + 4*x - 3*x^2) / (1 - x)^5.
a(n) = (n*(12 - 19*n + 6*n^2 + n^3)) / 12.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>4.
(End)

A326291 Number of unsortable factorizations 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, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 3, 0, 0, 0

Offset: 1

Gus Wiseman, Jun 24 2019

nonn

A factorization into factors > 1 is unsortable if there is no permutation (c_1,...,c_k) of the factors such that the maximum prime factor of c_i is at most the minimum prime factor of c_{i+1}. For example, the factorization (6*8*27) is sortable because the permutation (8,6,27) satisfies the condition.

			The a(180) = 10 unsortable factorizations:
  (2*3*3*10)  (5*6*6)   (3*60)
              (2*3*30)  (6*30)
              (2*9*10)  (9*20)
              (3*3*20)  (10*18)
              (3*6*10)
Missing from this list are:
  (2*2*3*3*5)  (2*2*5*9)   (4*5*9)   (2*90)   (180)
               (2*3*5*6)   (2*2*45)  (4*45)
               (3*3*4*5)   (2*5*18)  (5*36)
               (2*2*3*15)  (2*6*15)  (12*15)
                           (3*4*15)
                           (3*5*12)

Unsortable set partitions are A058681.
Unsortable normal multiset partitions are A326211.
MM-numbers of unsortable multiset partitions are A326258.
Cf. A000108, A001055, A001519, A016098, A112798, A302242, A324170, A324171.
Cf. A326209, A326212, A326243, A326256, A326257.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
lexsort[f_,c_]:=OrderedQ[PadRight[{f,c}]];
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[facs[n],!OrderedQ[Join@@Sort[primeMS/@#,lexsort]]&]],{n,100}]

A326277 Number of crossing normal multiset partitions of weight n.

Original entry on oeis.org

0, 0, 0, 0, 1, 22, 314, 3711, 39947

Offset: 0

Gus Wiseman, Jun 22 2019

A multiset partition is normal if it covers an initial interval of positive integers.

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.

			The a(5) = 22 crossing normal multiset partitions:
  {{1,3},{1,2,4}}  {{1},{1,3},{2,4}}
  {{1,3},{2,2,4}}  {{1},{2,4},{3,5}}
  {{1,3},{2,3,4}}  {{2},{1,3},{2,4}}
  {{1,3},{2,4,4}}  {{2},{1,4},{3,5}}
  {{1,3},{2,4,5}}  {{3},{1,3},{2,4}}
  {{1,4},{2,3,5}}  {{3},{1,4},{2,5}}
  {{2,4},{1,1,3}}  {{4},{1,3},{2,4}}
  {{2,4},{1,2,3}}  {{4},{1,3},{2,5}}
  {{2,4},{1,3,3}}  {{5},{1,3},{2,4}}
  {{2,4},{1,3,4}}
  {{2,4},{1,3,5}}
  {{2,5},{1,3,4}}
  {{3,5},{1,2,4}}

Crossing simple graphs are A326210.
Normal multiset partitions are A255906.
Non-crossing normal multiset partitions are A324171.
MM-numbers of crossing multiset partitions are A324170.
Cf. A000108, A016098, A054726, A058681.
Cf. A326211, A326212, A326255, A326256, A326258.

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]]]];
allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
croXQ[stn_]:=MatchQ[stn,{_,{_,x_,_,y_,_},_,{_,z_,_,t_,_},_}/;x

cp's OEIS Frontend

A326250 Number of weakly nesting simple graphs with vertices {1..n}.

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A326249 Number of capturing set partitions of {1..n} that are not nesting.

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A326259 MM-numbers of crossing, capturing multiset partitions (with empty parts allowed).

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A326279 Number of labeled n-vertex simple graphs containing either a crossing or a nesting pair of edges.

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A326260 MM-numbers of capturing, non-nesting multiset partitions (with empty parts allowed).

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A326247 Number of labeled n-vertex 2-edge multigraphs that are neither crossing nor nesting.

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A326291 Number of unsortable factorizations of n.

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A326277 Number of crossing normal multiset partitions of weight n.

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