A108305
Number of set partitions of {1, ..., n} that avoid 4-crossings.
Original entry on oeis.org
1, 1, 2, 5, 15, 52, 203, 877, 4139, 21119, 115495, 671969, 4132936, 26723063, 180775027, 1274056792, 9320514343, 70548979894, 550945607475, 4427978077331, 36544023687590, 309088822019071
Offset: 0
There are 4140 partitions of 8 elements, but a(8) = 4139 because the partition (1,5)(2,6)(3,7)(4,8) has a 4-crossing.
- M. Bousquet-Mélou and G. Xin, On partitions avoiding 3-crossings, arXiv:math/0506551 [math.CO], 2005-2006.
- Sophie Burrill, Sergi Elizalde, Marni Mishna and Lily Yen, A generating tree approach to k-nonnesting partitions and permutations, arXiv preprint arXiv:1108.5615 [math.CO], 2011.
- W. Chen, E. Deng, R. Du, R. Stanley, and C. Yan, Crossings and nestings of matchings and partitions, arXiv:math/0501230 [math.CO], 2005.
- Juan B. Gil and Jordan O. Tirrell, A simple bijection for classical and enhanced k-noncrossing partitions, arXiv:1806.09065 [math.CO], 2018. Also Discrete Mathematics (2019) Article 111705. doi:10.1016/j.disc.2019.111705
- M. Mishna and L. Yen, Set partitions with no k-nesting, arXiv:1106.5036 [math.CO], 2011-2012.
One more value from Burrill et al (2011). -
R. J. Mathar, May 25 2025
A192865
Number of set partitions of {1,...,n} that avoid enhanced 5-crossings (or 5-nestings).
Original entry on oeis.org
1, 1, 2, 5, 15, 52, 203, 877, 4140, 21146, 115945, 678012, 4205209, 27531954, 189486817, 1365888674, 10278272450, 80503198320, 654544093035, 5511256984436, 47950929125540
Offset: 0
There are 21147 partitions of 9 elements, but a(9)=21146 because the partition {1,9}{2,8}{3,7}{4, 6}{5} has an enhanced 5-nesting.
- M. Bousquet-Mélou and G. Xin, On partitions avoiding 3-crossings, math.CO/0506551.
- Sophie Burrill, Sergi Elizalde, Marni Mishna and Lily Yen, A generating tree approach to k-nonnesting partitions and permutations, arXiv preprint arXiv:1108.5615, 2011
- W. Chen, E. Deng, R. Du, R. P. Stanley, and C. Yan, Crossings and nestings of matchings and partitions, math.CO/0501230
- Juan B. Gil, Jordan O. Tirrell, A simple bijection for classical and enhanced k-noncrossing partitions, arXiv:1806.09065 [math.CO], 2018. Also Discrete Mathematics (2019) Article 111705. doi:10.1016/j.disc.2019.111705
A192866
Number of set partitions of {1, ..., n} that avoid enhanced 6-crossings (or enhanced 6-nestings).
Original entry on oeis.org
1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975, 678569, 4213555, 27643388, 190878823, 1382610179, 10474709625, 82784673008, 680933897225, 5816811952612, 51505026270176
Offset: 0
There are 678570 partitions of 11 elements, but a(11)=678569 because the partition {1,11}{2,10}{3,9}{4,8}{5,9}{6} has an enhanced 6-nesting.
- M. Bousquet-Mélou and G. Xin, On partitions avoiding 3-crossings, math.CO/0506551.
- Sophie Burrill, Sergi Elizalde, Marni Mishna and Lily Yen, A generating tree approach to k-nonnesting partitions and permutations, arXiv preprint arXiv:1108.5615, 2011
- W. Chen, E. Deng, R. Du, R. P. Stanley, and C. Yan, Crossings and nestings of matchings and partitions, math.CO/0501230
A192867
Number of set partitions of {1, ..., n} that avoid enhanced 7-crossings (or enhanced 7-nestings).
Original entry on oeis.org
1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975, 678570, 4213597, 27644436, 190899266, 1382956734, 10480097431, 82863928963, 682058946982, 5832425824171, 51718812364549
Offset: 0
There are 27644437 partitions of 13 elements, but a(13)=27644436 because the partition {1,13}{2,12}{3,11}{4,10}{5,9}{6,8} {7} has an enhanced 7-nesting.
- M. Bousquet-Mélou and G. Xin, On partitions avoiding 3-crossings, math.CO/0506551.
- Sophie Burrill, Sergi Elizalde, Marni Mishna and Lily Yen, A generating tree approach to k-nonnesting partitions and permutations, arXiv preprint arXiv:1108.5615, 2011
- W. Chen, E. Deng, R. Du, R. P. Stanley, and C. Yan, Crossings and nestings of matchings and partitions, math.CO/0501230
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