A100354 -id:A100354 - cp's OEIS frontend

A342864 Irregular table read by rows: T(n,k) is the number of permutations in S_n that have exactly k occurrences of the pattern 1432. 0 <= k <= A100354(n).

1, 1, 2, 6, 23, 1, 103, 11, 5, 0, 1, 513, 87, 68, 17, 18, 10, 0, 4, 2, 0, 1, 2761, 625, 626, 268, 274, 138, 112, 58, 51, 44, 31, 9, 15, 8, 12, 0, 5, 0, 0, 0, 3, 15767, 4378, 5038, 2781, 3060, 1697, 1817, 1036, 964, 773, 656, 450, 379, 320, 285, 148, 237, 97, 98, 55, 68, 61, 23, 30, 30, 13, 30, 0, 0, 0, 16, 0, 10, 0, 0, 1, 0, 0, 0, 0, 2

Offset: 0

Peter Kagey, Mar 26 2021

Equivalently the table for the patterns 2341, 3214, and 4123.

First column is A005802.

			Table begins:
  n\k|       0        1        2        3        4        5        6
  ---+-------------------------------------------------------------------
   0 |       1;
   1 |       1;
   2 |       2;
   3 |       6;
   4 |      23,       1;
   5 |     103,      11,       5,       0,       1;
   6 |     513,      87,      68,      17,      18,      10,       0, ...
   7 |    2761,     625,     626,     268,     274,     138,     112, ...
   8 |   15767,    4378,    5038,    2781,    3060,    1697,    1817, ...
   9 |   94359,   30671,   38541,   24731,   28881,   17943,   21193, ...
  10 |  586590,  216883,  289785,  205853,  251051,  170941,  211942, ...
  11 | 3763290, 1552588, 2172387, 1663964, 2096207, 1535129, 1954751, ...

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. A005802, A100354.

Analogous for other patterns: A008302 (12), A138159 (321), A263771 (312), A342840 (1342), A342860 (2413), A342861 (1324), A342862 (2143), A342863 (1243), A342865 (1234).

A342646 Maximal number of 4213 patterns in a permutation of 1,2,...,n.

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 6, 13, 24, 40, 62, 96, 138, 192, 264, 354

Offset: 0

Peter Kagey, Mar 20 2021

Equivalently the maximal number of 1342, 2431, and 3124 patterns.

			For n = 7, a(7) = 13 because the permutation 7532146 has 13 instances of the pattern 4213, namely: 7536, 7526, 7516, 7546, 7324, 7326, 7314, 7316, 7214, 7216, 5324, 5314, and 5214.
Moreover, all other permutations in S_7 have 13 or fewer instances of this pattern.

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).
David Bevan, The permutation class Av(4213,2143), arXiv:1510.06328 [math.CO], 2015.
FindStat, St000750: The number of occurrences of the pattern 4213 in a permutation.
Rob Pratt, Greatest number of occurrences of the pattern 4213 in a permutation, Mathematics Stack Exchange.
Eric Weisstein's World of Mathematics, Permutation Pattern

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

Cf. A005802, A022558

a(10)-a(12) from Rob Pratt

a(13)-a(15) from Bert Dobbelaere, Mar 26 2021

A100355 Maximal number of 15432 patterns in a permutation of 1,2,...,n.

Original entry on oeis.org

0, 0, 0, 0, 1, 5, 15, 35, 70, 140, 252, 420, 660, 990, 1485, 2145, 3003, 4095, 5460, 7280, 9520, 12240, 15504, 19381, 24226, 29926, 36576, 44276, 53135, 63761, 75905, 89705, 105305, 122865, 143340, 166272, 191850, 220270, 251755, 287715, 327395, 371043

Offset: 1

Vincent Vatter, Nov 18 2004

nonn

			a(24) = 19381; the 24-permutation with the most copies of 15432 is 1, 5, 4, 3, 2, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6.

W. Stromquist, Packing layered posets into posets, manuscript.

M. Bona, B. Sagan, and V. Vatter, Pattern frequency sequences and internal zeros, Advances in Applied Mathematics 28 (2002), pp. 395-420.
M. Hildebrand, B. Sagan, and V. Vatter, Bounding quantities related to the packing density of 1(l+1)l...2, Advances in Applied Mathematics, 33 (2004), pp. 633-653.

Cf. A061061, A100354, A100356.

a(n) = max(a(k) + k*choose(n-k, 4), 1 <= k < n)

A100356 Maximal number of 165432 patterns in a permutation of 1,2,...,n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 6, 21, 56, 126, 252, 504, 924, 1584, 2574, 4004, 6006, 9009, 13104, 18564, 25704, 34884, 46512, 62016, 81396, 105336, 134596, 170016, 212520, 265650, 328900, 403650, 491400, 593775, 712531, 855037, 1019467, 1208257, 1424017

Offset: 1

Vincent Vatter, Nov 18 2004

nonn

			a(12) = 502; the 12-permutation with the most copies of 165432 is 2, 1, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3.

W. Stromquist, Packing layered posets into posets, manuscript.

M. Bona, B. Sagan, and V. Vatter, Pattern frequency sequences and internal zeros, Advances in Applied Mathematics 28 (2002), pp. 395-420.
M. Hildebrand, B. Sagan, and V. Vatter, Bounding quantities related to the packing density of 1(l+1)l...2, Advances in Applied Mathematics, 33 (2004), pp. 633-653.

Cf. A061061, A100354, A100355.

a(n) = max(a(k) + k*choose(n-k, 5), 1 <= k < n)

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

A342864 Irregular table read by rows: T(n,k) is the number of permutations in S_n that have exactly k occurrences of the pattern 1432. 0 <= k <= A100354(n).

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A342646 Maximal number of 4213 patterns in a permutation of 1,2,...,n.

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A100355 Maximal number of 15432 patterns in a permutation of 1,2,...,n.

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A100356 Maximal number of 165432 patterns in a permutation of 1,2,...,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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