A342646 -id:A342646 - cp's OEIS frontend

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

Peter Kagey, Mar 24 2021

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!.

			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;

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

Cf. A022558, A342646.

Cf. A263771 (analogous for 312).

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 *)

A342853 Maximal number of 1324 patterns in a permutation of 1,2,...,n.

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 6, 13, 24, 42, 68, 106, 153, 217, 300

Offset: 0

Peter Kagey, Mar 25 2021

Equivalently the maximal number of 4231 patterns in a permutation of 1,2,...,n.

			For n = 5, the permutation 14325 has a(5) = 3 subsequences with the same relative order as 1324: 1435, 1425, and 1325.
All other permutations in S_5 have 3 or fewer such subsequences.

M. H. Albert, M. D. Atkinson, C. C.Handley, D. A. Holton, and W. Stromquist, On packing densities of permutations, The Electronic Journal of Combinatorics, 9(1) (2002).
Miklós Bóna. A new record for 1324-avoiding permutations European Journal of Mathematics 1 (2015), 198-206.
Anders Claesson, Vít Jelínek, and Einar Steingrímsson, Upper bounds for the Stanley-Wilf limit of 1324 and other layered patterns, Journal of Combinatorial Theory, Series A, 199.8 (2012), 1680-1691.
FindStat, St000405: The number of occurrences of the pattern 1324 in a permutation.
User Noodle9, answer to Patterns in Permutations, Code Golf Stack Exchange.
Eric Weisstein's World of Mathematics, Permutation Pattern

Analogous for other patterns: A000292 (123), A000332 (1234), A061061 (132), A100354 (1432), A342646 (4213), A342854 (2413).

a(11) from Code Golf Stack Exchange link added by Peter Kagey, Mar 25 2021

a(12)-a(14) from Hugo Pfoertner, Mar 26 2021

A342854 Maximal number of 2413 patterns in a permutation of 1,2,...,n.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 5, 9, 17, 26, 41, 60, 88, 120, 163, 213

Offset: 0

Peter Kagey, Mar 25 2021

Equivalently the maximal number of 3142 patterns in a permutation of 1,2,...,n.

			For n = 6, the permutation 246135 has a(6) = 5 subsequences with the same relative order as 2413: 2413, 2613, 2615, 4615, and 4635.
All other permutations in S_6 have 5 or fewer such subsequences.

M. H. Albert, M. D. Atkinson, C. C.Handley, D. A. Holton, and W. Stromquist, On packing densities of permutations, The Electronic Journal of Combinatorics, 9(1) (2002).
Eric Weisstein's World of Mathematics, Permutation Pattern

Analogous for other patterns: A000292 (123), A000332 (1234), A061061 (132), A100354 (1432), A342646 (4213), A342853 (1324).

Cf. A342860.

a(11)-a(14) from Hugo Pfoertner, Mar 26 2021

a(15) from Hugo Pfoertner, Apr 05 2021

cp's OEIS Frontend

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).

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A342853 Maximal number of 1324 patterns in a permutation of 1,2,...,n.

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A342854 Maximal number of 2413 patterns in a permutation of 1,2,...,n.

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