A326211 -id:A326211 - 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

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]]&]

A326255 MM-numbers of capturing multiset partitions.

Original entry on oeis.org

667, 989, 1334, 1633, 1769, 1817, 1978, 2001, 2021, 2323, 2461, 2599, 2623, 2668, 2967, 2987, 3197, 3266, 3335, 3538, 3634, 3713, 3749, 3956, 3979, 4002, 4042, 4163, 4171, 4331, 4379, 4429, 4439, 4577, 4646, 4669, 4747, 4819, 4859, 4899, 4922, 4945, 5029, 5198

Offset: 1

Gus Wiseman, Jun 20 2019

nonn

First differs from A326256 in having 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 capturing if it has two blocks of the form {...x...y...} and {...z...t...} where x < z and t < y or z < x and y < t. This 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:
   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}}
  2599: {{2,2},{1,2,3}}
  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}}

MM-numbers of crossing multiset partitions are A324170.
MM-numbers of nesting multiset partitions are A326256.
MM-numbers of crossing capturing multiset partitions are A326259.
Capturing set partitions are A326243.
Cf. A001055, A034827, A058681, A112798, A117662, A122880, A302242.
Cf. A326211, A326249, A054391, A326257, A326258.

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

A326248 Number of crossing, nesting set partitions of {1..n}.

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 28, 252, 1890, 13020, 86564, 571944, 3826230, 26233662, 185746860, 1364083084, 10410773076, 82609104802, 681130756224, 5829231836494, 51711093240518, 474821049202852, 4506533206814480, 44151320870760216, 445956292457725714

Offset: 0

Gus Wiseman, Jun 20 2019

nonn

A set 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, and nesting if it has two blocks of the form {...x,y...}, {...z,t...} where x < z < t < y or z < x < y < t.

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

Christian Sievers, Table of n, a(n) for n = 0..576

Crossing and nesting set partitions are (both) A016098.
Crossing, capturing set partitions are A326246.
Nesting, non-crossing set partitions are A122880.
Cf. A000108, A000110, A001519, A058681, A099947, A117662, A324170.
Cf. A326209, A326211, A326243, A326245, A326256, A326258.

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

a(n) = A000110(n) - 2*A000108(n) + A001519(n). - Christian Sievers, Oct 16 2024

a(11) and beyond from Christian Sievers, Oct 16 2024

A326212 Number of sortable normal multiset partitions of weight n.

Original entry on oeis.org

1, 1, 4, 15, 59, 230, 901, 3522, 13773, 53847, 210527, 823087, 3218002, 12581319, 49188823, 192312112, 751877137, 2939592383, 11492839729, 44933224559, 175674134309, 686828104551, 2685272063984, 10498530869151, 41045803846015, 160475597429847

Offset: 0

Gus Wiseman, Jun 19 2019

nonn

A multiset partition is normal if it covers an initial interval of positive integers. It is sortable if some 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 a(0) = 1 through a(3) = 15 multiset partitions:
  {}  {{1}}  {{1,1}}    {{1,1,1}}
             {{1,2}}    {{1,1,2}}
             {{1},{1}}  {{1,2,2}}
             {{1},{2}}  {{1,2,3}}
                        {{1},{1,1}}
                        {{1},{1,2}}
                        {{1,1},{2}}
                        {{1},{2,2}}
                        {{1,2},{2}}
                        {{1},{2,3}}
                        {{1,2},{3}}
                        {{1},{1},{1}}
                        {{1},{1},{2}}
                        {{1},{2},{2}}
                        {{1},{2},{3}}

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Sortable set partitions are A011782.
Unsortable normal multiset partitions are A326211.
Crossing normal multiset partitions are A326277.
Cf. A000041, A000108, A058681, A255906, A324170, A324171.
Cf. A326209, A326237, A326255, A326256, A326257, A326258.

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}]

seq(n) = my(p=1/eta(x + O(x*x^n))); Vec(((1 - x)*(1 - 2*x) - x^2*p)/(2*(1 - x)*(1 - 2*x) - (1 - 3*x + 4*x^2)*p)) \\ Andrew Howroyd, May 11 2023

A255906(n) = a(n) + A326211(n).

G.f.: ((1 - x)*(1 - 2*x) - x^2*P(x))/(2*(1 - x)*(1 - 2*x) - (1 - 3*x + 4*x^2)*P(x)) where P(x) is the g.f. of A000041. - Andrew Howroyd, May 11 2023

Terms a(10) and beyond from Andrew Howroyd, May 11 2023

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

Original entry on oeis.org

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).

A326257 MM-numbers of weakly nesting multiset partitions.

Original entry on oeis.org

49, 91, 98, 133, 147, 169, 182, 196, 203, 245, 247, 259, 266, 273, 294, 299, 301, 338, 343, 361, 364, 371, 377, 392, 399, 406, 427, 441, 455, 481, 490, 494, 497, 507, 518, 529, 532, 539, 546, 551, 553, 559, 588, 598, 602, 609, 623, 637, 665, 667, 676, 686, 689

Offset: 1

Gus Wiseman, Jun 21 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 weakly 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.

			The sequence of terms together with their multiset multisystems begins:
   49: {{1,1},{1,1}}
   91: {{1,1},{1,2}}
   98: {{},{1,1},{1,1}}
  133: {{1,1},{1,1,1}}
  147: {{1},{1,1},{1,1}}
  169: {{1,2},{1,2}}
  182: {{},{1,1},{1,2}}
  196: {{},{},{1,1},{1,1}}
  203: {{1,1},{1,3}}
  245: {{2},{1,1},{1,1}}
  247: {{1,2},{1,1,1}}
  259: {{1,1},{1,1,2}}
  266: {{},{1,1},{1,1,1}}
  273: {{1},{1,1},{1,2}}
  294: {{},{1},{1,1},{1,1}}
  299: {{1,2},{2,2}}
  301: {{1,1},{1,4}}
  338: {{},{1,2},{1,2}}
  343: {{1,1},{1,1},{1,1}}
  361: {{1,1,1},{1,1,1}}

MM-numbers of crossing multiset partitions are A324170.
MM-numbers of nesting multiset partitions are A324256.
MM-numbers of capturing multiset partitions are A326255.
Nesting set partitions are A016098.
Cf. A001055, A034827, A056239, A112798, A122880, A302242, A324171.
Cf. A326211, A326243, A326258, A326260.

wknXQ[stn_]:=MatchQ[stn,{_,{_,x_,y_,_},_,{_,z_,t_,_},_}/;(x<=z&&y>=t)||(x>=z&&y<=t)]
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],wknXQ[primeMS/@primeMS[#]]&]

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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A326256 MM-numbers of nesting multiset partitions.

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

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

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A326255 MM-numbers of capturing multiset partitions.

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A326248 Number of crossing, nesting set partitions of {1..n}.

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A326212 Number of sortable normal multiset partitions of weight n.

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