A342840 Irregular triangle: T(n,k) is the number of permutations in S_n that have exactly k occurrences of the pattern 4213. 0 <= k <= A342646(n).

1, 1, 2, 6, 23, 1, 103, 10, 6, 1, 512, 77, 69, 30, 21, 5, 6, 2740, 548, 598, 330, 335, 123, 174, 58, 58, 37, 26, 3, 9, 1, 15485, 3799, 4686, 2970, 3411, 1676, 2338, 1040, 1317, 878, 777, 363, 608, 230, 252, 165, 133, 30, 93, 26, 31, 4, 1, 3, 4, 91245, 26165, 35148, 24550, 30182, 17185, 24685, 12976, 16867, 12248, 12360, 7203, 11086, 5692, 6391, 5194, 5006, 2751, 3917, 2019, 2482, 1622, 1371, 812, 1233, 490, 495, 416, 360, 157, 282, 54, 78, 41, 29, 22, 49, 7, 4, 0, 6

Offset: 0

Author

Peter Kagey, Mar 24 2021

Keywords

• nonn
• tabf

Comments

The sequence is the same for the patterns 1342, 2431, and 3124.

The sequence appears to be the same for the patterns 1423, 2314, 3241, and 4132.

First column is given by A022558. Row sums given by n!.

Examples

			Triangle begins:
n\k |    0    1    2    3    4    5    6   7   8   9  10 11 12 13
----+-------------------------------------------------------------
  0 |    1;
  1 |    1;
  2 |    2;
  3 |    6;
  4 |   23,   1;
  5 |  103,  10,   6,   1;
  6 |  512,  77,  69,  30,  21,   5,   6;
  7 | 2740, 548, 598, 330, 335, 123, 174, 58, 58, 37, 26, 3, 9, 1;
		

Links

• Peter Kagey, Rows n = 0..13, flattened, based on Anders Kaseorg's Rust program at the Code Golf Stack Exchange link.
• FindStat, St000750: The number of occurrences of the pattern 4213 in a permutation.
• Anders Kaseorg, Answer: Patterns in Permutations, Code Golf Stack Exchange.
• Rob Pratt, Greatest number of occurrences of the pattern 4213 in a permutation, Mathematics Stack Exchange.
• Eric Weisstein's World of Mathematics, Permutation Pattern

Crossrefs

Cf. A022558, A342646.

Cf. A263771 (analogous for 312).

Programs

Mathematica

Join@@Array[Table[Length@Select[Permutations@Range@#,Length@Select[Subsets[#,{4}],Ordering@Ordering@#=={4,2,1,3}&]==k&],{k,0,Binomial[n+1,4]}]//.{a__,0}:>{a}&,8,0] (* Giorgos Kalogeropoulos, Mar 25 2021 *)