A342854 -id:A342854 - cp's OEIS frontend

A342860 Irregular table read by rows: T(n,k) is the number of permutations in S_n that have exactly k occurrences of the pattern 2413. 0 <= k <= A342854(n).

1, 1, 2, 6, 23, 1, 103, 9, 8, 512, 62, 82, 34, 28, 2, 2740, 402, 612, 384, 466, 94, 232, 42, 60, 8, 15485, 2593, 4187, 3036, 4356, 1746, 3132, 1064, 1918, 909, 654, 333, 612, 144, 104, 22, 24, 1, 91245, 16921, 28065, 21638, 33274, 17598, 31180, 12942, 24000, 14290, 15434, 7770, 15692, 5965, 6896, 3947, 5660, 2226, 3674, 1314, 1512, 516, 508, 204, 332, 37, 40

Offset: 0

Peter Kagey, Mar 26 2021

Equivalently the table for the pattern 3142.

First column is A022558.

			Triangle begins:
  n\k|       0       1        2        3        4        5        6
  ---+------------------------------------------------------------------
   0 |       1;
   1 |       1;
   2 |       2;
   3 |       6;
   4 |      23,      1;
   5 |     103,      9,       8;
   6 |     512,     62,      82,      34,      28,       2;
   7 |    2740,    402,     612,     384,     466,      94,     232, ...
   8 |   15485,   2593,    4187,    3036,    4356,    1746,    3132, ...
   9 |   91245,  16921,   28065,   21638,   33274,   17598,   31180, ...
  10 |  555662, 112196,  188514,  149946,  237128,  140954,  257686, ...
  11 | 3475090, 755920, 1278590, 1036826, 1658064, 1041598, 1933438, ...

Peter Kagey, Rows n = 0..13, flattened, based on Anders Kaseorg's Rust program in the Code Golf Stack Exchange link.
Anders Kaseorg, Answer: Patterns in Permutations, Code Golf Stack Exchange.

Cf. A022558, A342854.

Analogous for other patterns: A008302 (12), A138159 (321), A263771 (312), A342840 (1342), A342861 (1324), A342862 (2143), A342863 (1243), A342864 (1432), A342865 (1234).

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

cp's OEIS Frontend

A342860 Irregular table read by rows: T(n,k) is the number of permutations in S_n that have exactly k occurrences of the pattern 2413. 0 <= k <= A342854(n).

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