A326258 -id:A326258 - cp's OEIS frontend

A326211 Number of unsortable normal multiset partitions of weight n.

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).

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).

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

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[#]]&]

A326252 Number of digraphs with vertices {1..n} whose increasing edges are crossing.

Original entry on oeis.org

0, 0, 0, 0, 16384, 22020096, 62679678976, 556181084962816

Offset: 0

Gus Wiseman, Jun 30 2019

A directed edge (a,b) is increasing if a < b. Two edges (a,b), (c,d) are crossing if a < c < b < d or c < a < d < b.

Simple graphs whose edges are crossing are A326210.
Digraphs whose increasing edges are not crossing are A326251.
Digraphs whose edges are not crossing are A326237.
Cf. A002416, A002720, A229865, A324170, A326209, A326252, A326258.

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

a(n) = 2^(n * (n + 1)/2) * A326210(n).

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[#]]&]

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[#]]&]

cp's OEIS Frontend

A326211 Number of unsortable normal multiset partitions of weight 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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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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A326257 MM-numbers of weakly nesting multiset partitions.

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A326252 Number of digraphs with vertices {1..n} whose increasing edges are crossing.

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