A342865 -id:A342865 - 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).

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).

Original entry on oeis.org

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).

A342862 Irregular table read by rows: T(n,k) is the number of permutations in S_n that have exactly k occurrences of the pattern 2143. 0 <= k <= A028723(n + 1).

Original entry on oeis.org

1, 1, 2, 6, 23, 1, 103, 11, 4, 2, 513, 88, 53, 33, 18, 8, 6, 0, 0, 1, 2761, 642, 495, 340, 262, 160, 172, 65, 58, 39, 14, 6, 18, 0, 0, 6, 0, 0, 2, 15767, 4567, 4099, 3007, 2692, 1832, 2171, 1152, 1291, 968, 728, 457, 566, 174, 176, 221, 129, 14, 122, 29, 38, 52, 8, 0, 32, 9, 0, 10, 0, 0, 8, 0, 0, 0, 0, 0, 1

Offset: 0

Peter Kagey, Mar 26 2021

Equivalently the table for the pattern 3412.

First column is A005802.

			Triangle begins:
  n\k|       0        1        2        3        4        5        6
  ---+-------------------------------------------------------------------
   0 |       1;
   1 |       1;
   2 |       2;
   3 |       6;
   4 |      23,       1;
   5 |     103,      11,       4,       2;
   6 |     513,      88,      53,      33,      18,       8,       6, ...
   7 |    2761,     642,     495,     340,     262,     160,     172, ...
   8 |   15767,    4567,    4099,    3007,    2692,    1832,    2171, ...
   9 |   94359,   32443,   32345,   25049,   24492,   17732,   21841, ...
  10 |  586590,  232189,  250371,  203452,  211291,  160561,  201524, ...
  11 | 3763290, 1679295, 1926145, 1635315, 1776655, 1409304, 1787218, ...

Peter Kagey, Rows n = 0..13, flattened, based on Anders Kaseorg's Rust program in the Code Golf Stack Exchange link.
FindStat, St000407: The number of occurrences of the pattern 2143 in a permutation.
Anders Kaseorg, Answer: Patterns in Permutations, Code Golf Stack Exchange.

Cf. A005802, A028723.

Analogous for other patterns: A008302 (12), A138159 (321), A263771 (312), A342840 (1342), A342860 (2413), A342861 (1324), A342863 (1243), A342864 (1432), A342865 (1234).

A342863 Irregular table read by rows: T(n,k) is the number of permutations in S_n that have exactly k occurrences of the pattern 1243. 0 <= k <= A028723(n + 1).

Original entry on oeis.org

1, 1, 2, 6, 23, 1, 103, 11, 4, 2, 513, 88, 56, 32, 14, 7, 9, 0, 0, 1, 2761, 638, 543, 341, 235, 138, 173, 51, 42, 47, 34, 6, 17, 4, 0, 7, 1, 0, 2, 15767, 4478, 4600, 3119, 2658, 1710, 2180, 972, 975, 877, 771, 356, 542, 233, 184, 266, 157, 81, 130, 41, 60, 49, 16, 16, 37, 8, 9, 13, 3, 0, 10, 1, 0, 0, 0, 0, 1

Offset: 0

Peter Kagey, Mar 26 2021

Equivalently the table for the patterns 2134, 3421, and 4312.

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,       4,       2;
   6 |     513,      88,      56,      32,      14,       7,       9, ...
   7 |    2761,     638,     543,     341,     235,     138,     173, ...
   8 |   15767,    4478,    4600,    3119,    2658,    1710,    2180, ...
   9 |   94359,   31199,   36691,   26602,   25756,   17628,   22984, ...
  10 |  586590,  218033,  284370,  218957,  231390,  166338,  221429, ...
  11 | 3763290, 1535207, 2174352, 1767837, 1994176, 1496134, 2028316, ...

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, A028723.

Analogous for other patterns: A008302 (12), A138159 (321), A263771 (312), A342840 (1342), A342860 (2413), A342861 (1324), A342862 (2143), A342864 (1432), A342865 (1234).

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).

Original entry on oeis.org

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).

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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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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A342862 Irregular table read by rows: T(n,k) is the number of permutations in S_n that have exactly k occurrences of the pattern 2143. 0 <= k <= A028723(n + 1).

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A342863 Irregular table read by rows: T(n,k) is the number of permutations in S_n that have exactly k occurrences of the pattern 1243. 0 <= k <= A028723(n + 1).

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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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