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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.

%I A263771 #45 May 09 2021 10:14:27
%S A263771 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 A263771 22,10,0,2,429,330,464,410,526,394,475,365,360,298,281,175,206,126,93,
%U A263771 55,23,14,13,1,2,1430,1287,1950,1918,2593,2225,2858,2489,2682,2401
%N 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.
%C A263771 Row sums give A000142.
%C A263771 First column gives A000108.
%C A263771 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 A263771 Columns k=1-5 give: A002054(n-2) for n>=3, A082970, A082971, A138162, A138163. - _Alois P. Heinz_, Oct 27 2015
%H A263771 Alois P. Heinz, <a href="/A263771/b263771.txt">Rows n = 0..10, flattened</a>
%H A263771 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 A263771 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 A263771 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 A263771 Sum_{k>0} k * T(n,k) = A001810(n). - _Alois P. Heinz_, Oct 27 2015
%e A263771 Triangle begins:
%e A263771     1;
%e A263771     1;
%e A263771     2;
%e A263771     5,  1;
%e A263771    14,  5,   4,  1;
%e A263771    42, 21,  23, 14, 12,  5,  3;
%e A263771   132, 84, 107, 82, 96, 55, 64, 37, 29, 22, 10, 0, 2;
%e A263771   ...
%t A263771 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 A263771 Cf. A000108, A000142, A001810, A002054, A082970, A082971, A138162, A138159, A138163.
%K A263771 nonn,tabf
%O A263771 0,3
%A A263771 _Christian Stump_, Oct 26 2015
%E A263771 More terms from _Alois P. Heinz_, Oct 26 2015