cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Views

Author

Gus Wiseman, Jun 20 2019

Keywords

Comments

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.

Examples

			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}}

Crossrefs

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.

Programs

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

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

Views

Author

Gus Wiseman, Jun 21 2019

Keywords

Comments

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.

Examples

			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}}

Crossrefs

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.

Programs

  • Mathematica
    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[#]]&]

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

Views

Author

Gus Wiseman, Jun 20 2019

Keywords

Comments

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).

Examples

			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}}

Crossrefs

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.

Programs

  • Mathematica
    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
Showing 1-3 of 3 results.

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