A202058
Number of ascent sequences avoiding the pattern 000.
Original entry on oeis.org
1, 1, 2, 4, 10, 27, 83, 277, 1015, 4007, 17047, 77451, 374889, 1923168, 10427250, 59544957, 357236992, 2245822801, 14762969601, 101264286082, 723499803180, 5375063821727, 41459660565329, 331546282841906, 2745163969235517, 23505333233440927, 207895424692608432
Offset: 0
- Andrew Conway and Miles Conway, Table of n, a(n) for n = 0..176
- Andrew R. Conway, Miles Conway, Andrew Elvey Price and Anthony J. Guttmann, Pattern-avoiding ascent sequences of length 3, arXiv:2111.01279 [math.CO], Nov 01 2021.
- P. Duncan and Einar Steingrimsson, Pattern avoidance in ascent sequences, arXiv preprint arXiv:1109.3641, 2011
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b[n_, i_, t_, p_, k_] := b[n, i, t, p, k] = If[n==0, 1, Sum[If[Coefficient[ p, x, j]==k, 0, b[n-1, j, t + If[j>i, 1, 0], p+x^j, k]], {j, 1, t+1}]];
a[n_] := b[n, 0, 0, 0, Min[n, 2]];
Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 0, 17}] (* Jean-François Alcover, Sep 01 2018, after Alois P. Heinz in A294220 *)
A202062
Number of ascent sequences avoiding the pattern 201.
Original entry on oeis.org
1, 1, 2, 5, 15, 52, 201, 843, 3764, 17659, 86245, 435492, 2261769, 12033165, 65369590, 361661809, 2033429427, 11597912588, 67004252081, 391599609911, 2312726369640, 13789161819383, 82932744795049, 502777950712812, 3070529443569777, 18879637374473465, 116815588935673706, 727011479685559453
Offset: 0
- Andrew Conway and Miles Conway, Table of n, a(n) for n = 0..133
- Giulio Cerbai, Modified ascent sequences and Bell numbers, arXiv:2305.10820 [math.CO], 2023. See p. 27.
- Giulio Cerbai, Anders Claesson and Luca Ferrari, Stack sorting with restricted stacks, arXiv:1907.08142 [cs.DS], 2019.
- Andrew R. Conway, Miles Conway, Andrew Elvey Price and Anthony J. Guttmann, Pattern-avoiding ascent sequences of length 3, arXiv:2111.01279 [math.CO], Nov 01 2021.
- P. Duncan and Einar Steingrimsson, Pattern avoidance in ascent sequences, arXiv preprint arXiv:1109.3641 [math.CO], 2011.
- Anthony Guttmann and Vaclav Kotesovec, L-convex polyominoes and 201-avoiding ascent sequences, arXiv:2109.09928 [math.CO], 2021.
Total number of ascent sequences is given by
A022493.
A202059
Number of ascent sequences avoiding the pattern 100.
Original entry on oeis.org
1, 1, 2, 5, 14, 44, 153, 583, 2410, 10721, 50965, 257393, 1374187, 7722862, 45520064, 280502924, 1802060232, 12040040899, 83475921469, 599400745354, 4449689901306, 34096169966924, 269286884243138, 2189193150557825, 18297258191472880, 157049750065028868
Offset: 0
- Andrew Conway and Miles Conway, Table of n, a(n) for n = 0..628
- Andrew R. Conway, Miles Conway, Andrew Elvey Price and Anthony J. Guttmann, Pattern-avoiding ascent sequences of length 3, arXiv:2111.01279 [math.CO], Nov 01 2021.
- P. Duncan and Einar Steingrimsson, Pattern avoidance in ascent sequences, arXiv preprint arXiv:1109.3641, 2011
Corrected a(7), was 383, but should be 583 according to Duncan-Steimgrimsson paper and independent computation. -
Andrew Baxter, Jan 06 2014
A202060
Number of ascent sequences avoiding the pattern 110.
Original entry on oeis.org
1, 1, 2, 5, 14, 43, 143, 510, 1936, 7774, 32848, 145398, 671641, 3227218, 16084747, 82955090, 441773793, 2424845273, 13695855478, 79485625385, 473393639992, 2889930405750, 18064609329598, 115513453404597, 754956282308784, 5039064184597772, 34323984497482559
Offset: 0
- Andrew Conway and Miles Conway, Table of n, a(n) for n = 0..43
- Andrew R. Conway, Miles Conway, Andrew Elvey Price and Anthony J. Guttmann, Pattern-avoiding ascent sequences of length 3, arXiv:2111.01279 [math.CO], Nov 01 2021.
- P. Duncan and Einar Steingrimsson, Pattern avoidance in ascent sequences, arXiv preprint arXiv:1109.3641, 2011
A202061
Number of ascent sequences avoiding the pattern 120.
Original entry on oeis.org
1, 1, 2, 5, 14, 42, 133, 442, 1535, 5546, 20754, 80113, 317875, 1292648, 5374073, 22794182, 98462847, 432498659, 1929221610, 8728815103, 40017844229, 185727603829, 871897549029, 4137132922197, 19828476952117, 95934298966615, 468291607852143, 2305162065138433
Offset: 0
- Liang Chengwei, Shi Lecun and Cai Zhongyu, Table of n, a(n) for n = 0..500 (terms 0..74 from Andrew Conway and Miles Conway)
- Andrew R. Conway, Miles Conway, Andrew Elvey Price and Anthony J. Guttmann, Pattern-avoiding ascent sequences of length 3, arXiv:2111.01279 [math.CO], Nov 01 2021.
- Paul Duncan and Einar Steingrimsson, Pattern avoidance in ascent sequences, arXiv preprint arXiv:1109.3641 [math.CO], 2011.
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
A192126
Number of set partitions of {1, ..., n} that avoid 5-nestings.
Original entry on oeis.org
1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115974, 678530, 4212654, 27627153, 190624976, 1378972826, 10425400681, 82139435907, 672674215928, 5712423473216, 50193986895328, 455436027242590, 4259359394306331
Offset: 0
There are 115975 partitions of 10 elements, but a(10)=115974 because the partition {1,10}{2,9}{3,8}{4,7}{5,6} has a 5-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.
- 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.
A366774
Number of 2-distant 3-noncrossing partitions of {1,...,n}.
Original entry on oeis.org
1, 1, 2, 5, 15, 52, 203, 876, 4115, 20765, 111301, 627821, 3698873, 22623354, 142940629, 929208778, 6194162081, 42223649277, 293640007995, 2079196943605, 14964254850197, 109308213994757, 809340696014733, 6067405789245061, 46008536947670701, 352579939415882813
Offset: 0
There are 877 partitions of 7 elements, but a(7)=876 because the partition (1,5)(2,6)(3,7)(4) has a 2-distant 3-crossing.
- Juan B. Gil and Jordan O. Tirrell, A simple bijection for enhanced, classical, and 2-distant k-noncrossing partitions, Discrete Math. 343 (2020), no. 6, 111705, 5 pp.
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b[n_] := b[n] = If[n < 2, 1, (2*(5*n^2 + 12*n - 2)*b[n - 1] + 9*(-n^2 + n + 2)*b[n - 2])/((n + 4)*(n + 5))];
a[n_] := If[n == 0, 1, Sum[Binomial[n - 1, i]*b[i], {i, 0, n - 1}]];
Table[a[n], {n, 0, 200}] (* Jean-François Alcover, Nov 25 2023 *)
A192127
Number of set partitions of {1, ..., n} that avoid 6-nestings.
Original entry on oeis.org
1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975, 678570, 4213596, 27644383, 190897649, 1382919174, 10479355676, 82850735298, 681840170501, 5828967784989, 51665915664913, 473990899143781, 4493642492511044, 43959218211619150
Offset: 0
There are 4213597 partitions of 12 elements, but a(12)=4213597 because the partition {1,12}{2,11}{3,10}{4,9}{5,8}{6,7} has a 6-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.
- W. Chen, E. Deng, R. Du, R. Stanley, and C. Yan, Crossings and nestings of matchings and partitions, arXiv:math/0501230 [math.CO], 2005.
- M. Mishna and L. Yen, Set partitions with no k-nesting, arXiv:1106.5036 [math.CO], 2011-2012.
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