Brian Nakamura has authored 19 sequences. Here are the ten most recent ones:
A238987
Number of permutations of length n containing exactly 1 occurrence of the pattern 1324.
Original entry on oeis.org
0, 0, 0, 1, 10, 75, 522, 3579, 24670, 172198, 1219974, 8776255, 64082132, 474605417, 3562460562, 27079243352, 208281537572
Offset: 1
a(4)=1 since 1324 is the only length 4 permutation with 1 occurrence of the pattern 1324.
- B. K. Nakamura, Computational methods in permutation patterns, PhD Dissertation, Rutgers University, May 2013.
A224288
Number of permutations of length n containing exactly 2 occurrences of 123 and 2 occurrences of 132.
Original entry on oeis.org
0, 0, 0, 0, 1, 6, 26, 94, 306, 934, 2732, 7752, 21488, 58432, 156288, 411904, 1071104, 2750976, 6984704, 17545216, 43634688, 107511808, 262602752, 636223488, 1529741312, 3652059136, 8660975616, 20412104704, 47826599936, 111446851584, 258360737792, 596044152832
Offset: 0
a(4) = 1: (1,2,4,3).
a(5) = 6: (2,3,5,1,4), (2,3,5,4,1), (2,5,1,3,4), (3,1,4,5,2), (4,1,2,5,3), (5,1,2,4,3).
- Alois P. Heinz, Table of n, a(n) for n = 0..1000
- B. Nakamura, Approaches for enumerating permutations with a prescribed number of occurrences of patterns, arXiv 1301.5080 [math.CO], 2013.
- B. Nakamura, A Maple package for enumerating n-permutations with r occurrences of the pattern 123 and s occurrences of the pattern 132 [Broken link]
- Index entries for linear recurrences with constant coefficients, signature (10,-40,80,-80,32).
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# Programs can be obtained from the Nakamura link
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Join[{0, 0, 0, 0, 1}, LinearRecurrence[{10, -40, 80, -80, 32}, {6, 26, 94, 306, 934}, 27]] (* Jean-François Alcover, Feb 29 2020 *)
A224298
Number of permutations in S_n containing exactly 3 increasing subsequences of length 4.
Original entry on oeis.org
0, 0, 0, 0, 0, 10, 196, 2477, 25886, 244233, 2167834, 18510734, 154082218, 1260811144, 10198142484, 81848366557, 653537296202, 5201485318177, 41321901094750, 327996498249202
Offset: 1
- B. Nakamura and D. Zeilberger, Using Noonan-Zeilberger Functional Equations to enumerate (in Polynomial Time!) Generalized Wilf classes, Adv. in Appl. Math. 50 (2013), 356-366.
A224297
Number of permutations of length n containing exactly 1 occurrence of 12345 and 1 occurrence of 12354.
Original entry on oeis.org
0, 0, 0, 0, 0, 0, 2, 56, 984, 14004, 177140, 2083560, 23374060, 254007320, 2700941710, 28293706782, 293369952474, 3020937207142, 30967869359950, 316581042622444, 3231631411510936, 32971319565628464, 336462340456201700
Offset: 1
A224296
Number of permutations of length n avoiding 12345 and containing exactly 1 occurrence of 12354.
Original entry on oeis.org
0, 0, 0, 0, 1, 18, 219, 2294, 22457, 212994, 1992364, 18553588, 172889838, 1616683912, 15193902185, 143632773054, 1366274116893, 13078787860290, 125984999505040, 1221032658065388, 11904323953163434, 116721427446951748, 1150690705785803998
Offset: 1
A224295
Number of permutations of length n avoiding 12345 and 12354.
Original entry on oeis.org
1, 1, 2, 6, 24, 118, 672, 4256, 29176, 212586, 1625704, 12930160, 106242392, 897210996, 7756325952, 68422701792, 614341492144, 5602330498170, 51798365474872, 484856381630288, 4589003801130456, 43870126242653020, 423219224419273888, 4116816114087389056, 40351014094161799568, 398270701521760650532
Offset: 0
- Jay Pantone, Table of n, a(n) for n = 0..790
- Michael H. Albert, Christian Bean, Anders Claesson, Émile Nadeau, Jay Pantone, and Henning Ulfarsson, Combinatorial Exploration: An algorithmic framework for enumeration, arXiv:2202.07715 [math.CO], 2022.
- Michael H. Albert, Christian Bean, Anders Claesson, Émile Nadeau, Jay Pantone, and Henning Ulfarsson, PermPAL Database
- Christian Bean, Émile Nadeau, Jay Pantone, and Henning Ulfarsson, Permutations avoiding bipartite partially ordered patterns have a regular insertion encoding, The Electronic Journal of Combinatorics, Volume 31, Issue 3 (2024); arXiv preprint, arXiv:2312.07716 [math.CO], 2023.
- Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, arXiv:1903.08946 [math.CO], 2019.
- Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, The Electronic Journal of Combinatorics 26(3) (2019), P3.26.
- B. Nakamura, Approaches for enumerating permutations with a prescribed number of occurrences of patterns, arXiv 1301.5080 [math.CO], 2013.
A224294
Number of permutations of length n containing exactly 2 occurrences of 1234 and 2 occurrences of 1243.
Original entry on oeis.org
0, 0, 0, 0, 0, 2, 34, 382, 3518, 28708, 216098, 1536960, 10488708, 69393216, 448301130, 2842611306, 17758399872, 109613418746, 669950441982, 4061411038376, 24453911859134, 146393798673526, 872121958672162, 5173931524573894, 30584965562156650
Offset: 1
A224293
Number of permutations of length n containing exactly 1 occurrence of 1234 and 1 occurrence of 1243.
Original entry on oeis.org
0, 0, 0, 0, 0, 2, 32, 322, 2634, 19216, 130662, 848284, 5334332, 32788726, 198201268, 1183152686, 6995479542, 41055940756, 239560353258, 1391435203864, 8052418072728, 46464481278410, 267483032264168, 1536914425518970, 8817393642571330
Offset: 1
A224292
Number of permutations of length n avoiding 1234 and containing exactly 1 occurrence of 1243.
Original entry on oeis.org
0, 0, 0, 1, 10, 71, 444, 2617, 14958, 84063, 467960, 2591265, 14308722, 78911943, 435066228, 2399404345, 13242035030, 73150006271, 404525810928, 2239684086529, 12415205896794, 68905402034311, 382897019146540, 2130255439081721, 11865617032955070
Offset: 1
A224291
Number of permutations of length n containing exactly 4 occurrences of 123 and 4 occurrences of 132.
Original entry on oeis.org
0, 0, 0, 0, 1, 11, 60, 270, 1084, 4028, 14144, 47577, 154740, 489728, 1514786, 4593118, 13682374, 40106060, 115824376, 329901232, 927585696, 2576685888, 7076644480, 19228648192, 51725149184
Offset: 1
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