A263771 Triangle read by rows: T(n,k) (n>=0, k>=0) is the number of permutations of n and k occurrences of the pattern 312.

1, 1, 2, 5, 1, 14, 5, 4, 1, 42, 21, 23, 14, 12, 5, 3, 132, 84, 107, 82, 96, 55, 64, 37, 29, 22, 10, 0, 2, 429, 330, 464, 410, 526, 394, 475, 365, 360, 298, 281, 175, 206, 126, 93, 55, 23, 14, 13, 1, 2, 1430, 1287, 1950, 1918, 2593, 2225, 2858, 2489, 2682, 2401

Offset

0,3

Comments

Row sums give A000142.

First column gives A000108.

Also the number of permutations of n and k occurrences of either of the fixed pattern 132, 213, 231 (these are all connected by reverses and inverses).

Columns k=1-5 give: A002054(n-2) for n>=3, A082970, A082971, A138162, A138163. - Alois P. Heinz, Oct 27 2015

Links

Alois P. Heinz, Rows n = 0..10, flattened

Miklós Bóna, The Number of Permutations with Exactly r 132-Subsequences Is P-Recursive in the Size!, Advances in Applied Mathematics, Volume 18, Issue 4, May 1997, Pages 510-522.

FindStat - Combinatorial Statistic Finder, The number of occurrences of the pattern 312 in a permutation, The number of occurrences of the pattern 213 in a permutation, The number of occurrences of the pattern 231 in a permutation, The number of occurrences of the pattern 132 in a permutation

T. Mansour and A. Vainshtein, Counting occurrences of 132 in a permutation, arXiv:math/0105073 [math.CO], 2001.

Formula

Sum_{k>0} k * T(n,k) = A001810(n). - Alois P. Heinz, Oct 27 2015

Example

Triangle begins:
    1;
    1;
    2;
    5,  1;
   14,  5,   4,  1;
   42, 21,  23, 14, 12,  5,  3;
  132, 84, 107, 82, 96, 55, 64, 37, 29, 22, 10, 0, 2;
  ...

Mathematica

Join@@Array[Table[Length@Select[Permutations@Range@#, Length@Select[Subsets[#, {3}], Ordering@Ordering@#=={3, 1, 2}&]==k&], {k, 0, Binomial[#+1, 3]}]//.{a__, 0}:>{a}&, 8, 0]  (* Giorgos Kalogeropoulos, Mar 26 2021 *)

Crossrefs

Cf. A000108, A000142, A001810, A002054, A082970, A082971, A138162, A138159, A138163.

Sequence in context: A101920 A114494 A118964 * A073187 A138159 A118919

Adjacent sequences: A263768 A263769 A263770 * A263772 A263773 A263774

Keyword

nonn,tabf

Author

Christian Stump, Oct 26 2015

Extensions

More terms from Alois P. Heinz, Oct 26 2015

Status

approved