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
Examples
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;
...
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.
Programs
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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 *)
Formula
Sum_{k>0} k * T(n,k) = A001810(n). - Alois P. Heinz, Oct 27 2015
Extensions
More terms from Alois P. Heinz, Oct 26 2015
Comments