A279571
Number of length n inversion sequences avoiding the patterns 100, 101, and 201.
Original entry on oeis.org
1, 1, 2, 6, 22, 92, 424, 2106, 11102, 61436, 353980, 2110366, 12955020, 81569168, 525106698, 3447244188, 23028080268, 156246994264, 1075127143948, 7492458675666, 52820934349420, 376331681648402, 2707312468516446, 19650530699752470, 143807774782994412, 1060472244838174574, 7875713244761349666, 58876660310205135380, 442862775457168812898, 3350397169412102710198
Offset: 0
The length 4 inversion sequences avoiding (100,101,201) are 0000, 0001, 0002, 0003, 0010, 0011, 0012, 0013, 0020, 0021, 0022, 0023, 0102, 0103, 0110, 0111, 0112, 0113, 0120, 0121, 0122, 0123.
Cf.
A000108,
A057552,
A108307,
A117106,
A263777,
A263778,
A263779,
A263780,
A279551,
A279552,
A279553,
A279554,
A279555,
A279556,
A279557,
A279558,
A279559,
A279560,
A279561,
A279562,
A279563,
A279564,
A279565,
A279566,
A279567,
A279568,
A279569,
A279570,
A279572,
A279573.
-
b:= proc(n, i, s, m) option remember;
`if`(n=0, 1, add(b(n-1, i+1, s minus {$j..m-
`if`(j=m, 1, 0)} union {i+1}, max(m, j)), j=s))
end:
a:= n-> b(n, 1, {1}, 0):
seq(a(n), n=0..15); # Alois P. Heinz, Feb 22 2017
-
b[n_, i_, s_, m_] := b[n, i, s, m] = If[n == 0, 1, Sum[b[n-1, i+1, s ~Complement~ Range[j, m - If[j == m, 1, 0]] ~Union~ {i+1}, Max[m, j]], {j, s}]];
a[n_] := b[n, 1, {1}, 0];
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Oct 27 2017, after Alois P. Heinz *)
A192855
Number of set partitions of {1, ..., n} that avoid enhanced 4-crossings (or enhanced 4-nestings).
Original entry on oeis.org
1, 1, 2, 5, 15, 52, 203, 876, 4120, 20883, 113034, 648410, 3917021, 24785452, 163525976, 1120523114, 7947399981, 58172358642, 438300848329, 3391585460591, 26898763482122
Offset: 0
There are 877 partitions of 7 elements, but a(7)=51 because the partition {1,7}{2,6}{3,5}{4} has an enhanced 4-nesting.
- 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], 2011-2014.
- W. Chen, E. Deng, R. Du, R. P. 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], 2018-2023. Also Discrete Mathematics (2019) Article 111705, doi:10.1016/j.disc.2019.111705.
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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