A342853 -id:A342853 - cp's OEIS frontend

A342861 Irregular table read by rows: T(n,k) is the number of permutations in S_n that have exactly k occurrences of the pattern 1324. 0 <= k <= A342853(n).

1, 1, 2, 6, 23, 1, 103, 10, 6, 1, 513, 75, 74, 26, 17, 9, 6, 2762, 522, 645, 321, 290, 130, 166, 47, 54, 48, 41, 4, 8, 2, 15793, 3579, 5023, 3058, 3232, 1527, 2228, 874, 1159, 893, 875, 340, 503, 281, 269, 207, 156, 112, 123, 21, 54, 2, 0, 6, 5

Offset: 0

Peter Kagey, Mar 26 2021

Equivalently the table for the pattern 4231.

First column is A061552.

			Triangle begins:
  n\k|       0        1        2        3        4        5        6
  ---+-------------------------------------------------------------------
   0 |       1;
   1 |       1;
   2 |       2;
   3 |       6;
   4 |      23,       1;
   5 |     103,      10,       6,       1;
   6 |     513,      75,      74,      26,      17,       9,       6;
   7 |    2762,     522,     645,     321,     290,     130,     166, ...
   8 |   15793,    3579,    5023,    3058,    3232,    1527,    2228, ...
   9 |   94776,   24670,   37549,   26174,   30409,   15966,   23762, ...
  10 |  591950,  172198,  277089,  213122,  264667,  154452,  228665, ...
  11 | 3824112, 1219974, 2043416, 1693787, 2213548, 1420513, 2086877, ...

Peter Kagey, Rows n = 0..13, flattened, based on Anders Kaseorg's Rust program in the Code Golf Stack Exchange link.
FindStat St000405: The number of occurrences of the pattern 1324 in a permutation.
Anders Kaseorg, Answer: Patterns in Permutations, Code Golf Stack Exchange.

Cf. A061552, A342853.

Analogous for other patterns: A008302 (12), A138159 (321), A263771 (312), A342840 (1342), A342860 (2413), A342862 (2143), A342863 (1243), A342864 (1432), A342865 (1234).

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

A342861 Irregular table read by rows: T(n,k) is the number of permutations in S_n that have exactly k occurrences of the pattern 1324. 0 <= k <= A342853(n).

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