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 *)
A224290
Number of permutations of length n containing exactly 3 occurrences of 123 and 3 occurrences of 132.
Original entry on oeis.org
0, 0, 0, 0, 0, 1, 6, 30, 136, 566, 2176, 7808, 26440, 85332, 264632, 793792, 2315136, 6592640, 18390784, 50392064, 135921664, 361536512, 949708800, 2466807808, 6342115328, 16153509888, 40790523904, 102186352640, 254105092096, 627533152256, 1539764125696
Offset: 0
a(5) = 1: (1,4,3,2,5).
a(6) = 6: (2,5,4,3,1,6), (2,5,4,3,6,1), (3,5,1,4,6,2), (3,6,1,4,2,5), (5,1,4,3,2,6), (6,1,4,3,2,5).
- 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 (14,-84,280,-560,672,-448,128).
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# Programs can be obtained from the Nakamura link
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Join[{0, 0, 0, 0, 0, 1, 6}, LinearRecurrence[{14, -84, 280, -560, 672, -448, 128}, {30, 136, 566, 2176, 7808, 26440, 85332}, 33]] (* Jean-François Alcover, Nov 28 2018 *)
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concat([0,0,0,0,0], Vec(x^5*(1 - 8*x + 30*x^2 - 60*x^3 + 62*x^4 - 36*x^5 + 24*x^6 - 8*x^7 + 4*x^8) / (1 - 2*x)^7 + O(x^40))) \\ Colin Barker, Nov 28 2018
A224289
Number of permutations of length n containing exactly 1 occurrence of 123 and 2 occurrences of 132.
Original entry on oeis.org
0, 0, 0, 2, 8, 26, 79, 232, 664, 1856, 5072, 13568, 35584, 91648, 232192, 579584, 1427456, 3473408, 8359936, 19922944, 47054848, 110231552, 256311296, 591921152, 1358430208, 3099590656, 7034896384, 15888023552, 35718692864, 79960211456, 178291474432, 396076515328, 876844417024
Offset: 1
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# Programs can be obtained from author's personal website.
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LinearRecurrence[{8,-24,32,-16},{0,0,0,2,8,26,79},40] (* Harvey P. Dale, Jun 23 2017 *)
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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