A263771 - OEIS

%I #45 May 09 2021 10:14:27

%S 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,

%T 22,10,0,2,429,330,464,410,526,394,475,365,360,298,281,175,206,126,93,

%U 55,23,14,13,1,2,1430,1287,1950,1918,2593,2225,2858,2489,2682,2401

%N 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.

%C Row sums give A000142.

%C First column gives A000108.

%C 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).

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

%H Alois P. Heinz, <a href="/A263771/b263771.txt">Rows n = 0..10, flattened</a>

%H Miklós Bóna, <a href="http://dx.doi.org/10.1006/aama.1997.0528">The Number of Permutations with Exactly r 132-Subsequences Is P-Recursive in the Size!</a>, Advances in Applied Mathematics, Volume 18, Issue 4, May 1997, Pages 510-522.

%H FindStat - Combinatorial Statistic Finder, <a href="http://www.findstat.org/StatisticsDatabase/St000217">The number of occurrences of the pattern 312 in a permutation</a>, <a href="http://www.findstat.org/StatisticsDatabase/St000218">The number of occurrences of the pattern 213 in a permutation</a>, <a href="http://www.findstat.org/StatisticsDatabase/St000219">The number of occurrences of the pattern 231 in a permutation</a>, <a href="http://www.findstat.org/StatisticsDatabase/St000220">The number of occurrences of the pattern 132 in a permutation</a>

%H T. Mansour and A. Vainshtein, <a href="https://arxiv.org/abs/math/0105073">Counting occurrences of 132 in a permutation</a>, arXiv:math/0105073 [math.CO], 2001.

%F Sum_{k>0} k * T(n,k) = A001810(n). - _Alois P. Heinz_, Oct 27 2015

%e Triangle begins:

%e     1;

%e     1;

%e     2;

%e     5,  1;

%e    14,  5,   4,  1;

%e    42, 21,  23, 14, 12,  5,  3;

%e   132, 84, 107, 82, 96, 55, 64, 37, 29, 22, 10, 0, 2;

%e   ...

%t 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 *)

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

%K nonn,tabf

%O 0,3

%A _Christian Stump_, Oct 26 2015

%E More terms from _Alois P. Heinz_, Oct 26 2015