A328500
Number of inversion sequences of length n avoiding the consecutive pattern 102.
Original entry on oeis.org
1, 1, 2, 6, 22, 96, 492, 2902, 19350, 143918, 1181540, 10614698, 103589738, 1091367634, 12346368424, 149276823258, 1921099070062, 26220186000950, 378308908684300, 5753387612678314, 91988260677198002, 1542570178562361018, 27072325866355742048
Offset: 0
- Alois P. Heinz, Table of n, a(n) for n = 0..250
- Juan S. Auli, Pattern Avoidance in Inversion Sequences, Ph. D. thesis, Dartmouth College, ProQuest Dissertations Publishing (2020), 27964164.
- Juan S. Auli, Sergi Elizalde, Consecutive Patterns in Inversion Sequences, arXiv:1904.02694 [math.CO], 2019. See Table 3.
-
b:= proc(n, j, t) option remember; `if`(n=0, 1, add(
`if`(i<=j or i>=t, b(n-1, i, j), 0), i=1..n))
end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..25); # Alois P. Heinz, Oct 18 2019
-
b[n_, j_, t_] := b[n, j, t] = If[n == 0, 1, Sum[If[i <= j || i >= t, b[n - 1, i, j], 0], {i, 1, n}]];
a[n_] := b[n, 0, 0];
a /@ Range[0, 25] (* Jean-François Alcover, Mar 01 2020, after Alois P. Heinz *)
A328501
Number of inversion sequences of length n avoiding the consecutive pattern 201.
Original entry on oeis.org
1, 1, 2, 6, 24, 118, 684, 4548, 34036, 282696, 2577936, 25589100, 274539856, 3164909164, 39006958856, 511759353776, 7120140764224, 104703385864788, 1622530610142744, 26425922582118000, 451264786489454168, 8062192403534869432, 150395837509736576208
Offset: 0
- Vaclav Kotesovec, Table of n, a(n) for n = 0..465
- Juan S. Auli, Pattern Avoidance in Inversion Sequences, Ph. D. thesis, Dartmouth College, ProQuest Dissertations Publishing (2020), 27964164.
- Juan S. Auli, Sergi Elizalde, Consecutive Patterns in Inversion Sequences, arXiv:1904.02694 [math.CO], 2019. See Table 3.
A328507
Number of inversion sequences of length n avoiding the consecutive pattern 101.
Original entry on oeis.org
1, 1, 2, 6, 23, 109, 619, 4113, 31352, 269841, 2589026, 27404677, 317265161, 3988181568, 54099618419, 787705115000, 12253696410675, 202831037178017, 3559585021719875, 66018657264425355, 1290284788431977106, 26505045303122642171, 570918508059059670322
Offset: 0
- Alois P. Heinz, Table of n, a(n) for n = 0..250
- Juan S. Auli, Pattern Avoidance in Inversion Sequences, Ph. D. thesis, Dartmouth College, ProQuest Dissertations Publishing (2020), 27964164.
- Juan S. Auli, Sergi Elizalde, Consecutive Patterns in Inversion Sequences, arXiv:1904.02694 [math.CO], 2019. See Table 4.
-
b:= proc(n, j, t) option remember; `if`(n=0, 1, add(
`if`(i<=j or i<>t, b(n-1, i, j), 0), i=1..n))
end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..25); # Alois P. Heinz, Oct 18 2019
-
b[n_, j_, t_] := b[n, j, t] = If[n == 0, 1, Sum[If[i <= j || i != t, b[n-1, i, j], 0], {i, 1, n}]];
a[n_] := b[n, 0, 0];
a /@ Range[0, 25] (* Jean-François Alcover, Mar 19 2020, after Alois P. Heinz *)
A324131
Number of permutations of [n] that avoid the shuffle pattern s-k-t, where s = 1 and t = 123.
Original entry on oeis.org
1, 1, 2, 6, 24, 116, 657, 4260, 31144, 253400, 2271250, 22234380, 236042879, 2700973070, 33139335352, 433996381926, 6042468288640, 89124117755852, 1388234052651161, 22771513253008320, 392354340340237176, 7084700602143004688, 133785708212530414358, 2636998678988431607188
Offset: 0
From _Petros Hadjicostas_, Nov 01 2019: (Start)
In a permutation of [n] that contains the shuffle pattern s-k-t, where s = 1 and t = 123, k should be greater than the numbers in pattern s and the numbers in pattern t. (The numbers in each of the patterns s and t should be contiguous.) Clearly, for n = 0..4, all permutations of [n] avoid this shuffle pattern (since we need at least five numbers to get this pattern). Hence, a(n) = n! for n = 0..4.
For n = 5, the permutations of [n] that contain this shuffle pattern should have k = 5 and the last three numbers in these permutations (with pattern t) should be one of the choices 123, 124, 134, and 234. The corresponding permutations that contain this shuffle pattern are 45123, 35124, 25134, and 15234. Hence a(5) = 5! - 4 = 116. (End)
A324135
Number of permutations of [n] that avoid the shuffle pattern s-k-t, where s = 12 and t = 123.
Original entry on oeis.org
1, 1, 2, 6, 24, 120, 710, 4815, 36650, 308778, 2850294, 28602468, 310041806, 3610879857, 44975227466, 596677473990, 8401332033264, 125140942951896, 1966223504686334, 32501786913873447, 563877339150924866, 10245134152041643818, 194553155073687332550, 3854328529787275833204
Offset: 0
A324137
Number of permutations of [n] that avoid the shuffle pattern s-k-t, where s = 123 and t = 123.
Original entry on oeis.org
1, 1, 2, 6, 24, 120, 720, 5020, 39790, 352470, 3445032, 36775404, 425282892, 5292245764, 70471602994, 999394962306, 15032677450752, 238984379214960, 4002966472631160, 70448661940661068, 1299478739106621670, 25067231918730741438, 504674373639695198712, 10584965637367018566180
Offset: 0
A324138
Number of permutations of [n] that avoid the shuffle pattern s-k-t, where s = 123 and t = 132.
Original entry on oeis.org
1, 1, 2, 6, 24, 120, 720, 5020, 39755, 351518, 3425572, 36419844, 419026188, 5182797757, 68535001302, 964404124479, 14383519018582, 226579159065496, 3758349089828472, 65466833442028670, 1194655878120996337, 22788580047064423474, 453513206778006345040
Offset: 0
From _Petros Hadjicostas_, Nov 01 2019: (Start)
In a permutation of [n] that contains the shuffle pattern s-k-t, where s = 123 and t = 132, k should be greater than the numbers in pattern s and the numbers in pattern t. (The numbers in each of the patterns s and t should be contiguous.) Clearly, for n = 0..6, all permutations of [n] avoid this shuffle pattern (since we need at least seven numbers to get this pattern). Hence, a(n) = n! for n = 0..6.
For n = 7, k should be equal to 7, and for the pattern s = 123 we have binomial(6,3) = 20 choices: 123, 124, 125, ..., 456. The corresponding permutations of [7] that contain this shuffle pattern are 1237465, 1247365, 1257364, ..., 4567132. Thus, a(7) = 7! - 20 = 5020. (End)