A138159 -id:A138159 - cp's OEIS frontend

A056986 Number of permutations on {1,...,n} containing any given pattern alpha in the symmetric group S_3.

0, 0, 1, 10, 78, 588, 4611, 38890, 358018, 3612004, 39858014, 478793588, 6226277900, 87175616760, 1307664673155, 20922754530330, 355687298451210, 6402373228089300, 121645098641568810, 2432902001612519580, 51090942147243172980, 1124000727686125116360

Offset: 1

Eric W. Weisstein

This is well-defined because for all patterns alpha in S_3 the number of permutations in S_n avoiding alpha is the same (the Catalan numbers). - Emeric Deutsch, May 05 2008

			a(4) = 10 because, taking, for example, the pattern alpha=321, we have 3214, 3241, 1432, 2431, 3421, 4213, 4132, 4231, 4312 and 4321.

Alois P. Heinz, Table of n, a(n) for n = 1..170
FindStat - Combinatorial Statistic Finder, The number of occurrences of the pattern [1,2,3] inside a permutation of length at least 3, The number of occurrences of the pattern [1,3,2] in a permutation, The number of occurrences of the pattern [2,1,3] in a permutation, The number of occurrences of the pattern [2,3,1] in a permutation, The number of occurrences of the pattern [3,1,2] in a permutation
R. Simion and F. W. Schmidt, Restricted permutations, European J. Combin., 6, pp. 383-406, 1985.
Eric Weisstein's World of Mathematics, Permutation Pattern

Cf. A000108, A000142, A138159, A214015, A214152.

A056986:= func< n | Factorial(n) - Catalan(n) >;
[A056986(n): n in [1..30]]; // G. C. Greubel, Oct 06 2024

a:= n-> n! -binomial(2*n, n)/(n+1):
seq(a(n), n=1..25);  # Alois P. Heinz, Jul 05 2012

Mathematica
```
Table[n! -CatalanNumber[n], {n,30}]
```

a(n)=n!-binomial(n+n,n+1)/n \\ Charles R Greathouse IV, Jun 10 2011

def A056986(n): return factorial(n) - catalan_number(n)
[A056986(n) for n in range(1,31)] # G. C. Greubel, Oct 06 2024

From Alois P. Heinz, Jul 05 2012: (Start)
a(n) = A214152(n, 3).
a(n) = A000142(n) - A000108(n).
a(n) = A000142(n) - A214015(n, 2). (End)
E.g.f.: 1/(1 - x) - exp(2*x)*(BesselI(0,2*x) - BesselI(1,2*x)). - Ilya Gutkovskiy, Jan 21 2017

A263771 Triangle read by rows: T(n,k) (n>=0, k>=0) is the number of permutations of n and k occurrences of the pattern 312.

Original entry on oeis.org

1, 1, 2, 5, 1, 14, 5, 4, 1, 42, 21, 23, 14, 12, 5, 3, 132, 84, 107, 82, 96, 55, 64, 37, 29, 22, 10, 0, 2, 429, 330, 464, 410, 526, 394, 475, 365, 360, 298, 281, 175, 206, 126, 93, 55, 23, 14, 13, 1, 2, 1430, 1287, 1950, 1918, 2593, 2225, 2858, 2489, 2682, 2401

Offset: 0

Christian Stump, Oct 26 2015

Row sums give A000142.
First column gives A000108.
Also the number of permutations of n and k occurrences of either of the fixed pattern 132, 213, 231 (these are all connected by reverses and inverses).
Columns k=1-5 give: A002054(n-2) for n>=3, A082970, A082971, A138162, A138163. - Alois P. Heinz, Oct 27 2015

			Triangle begins:
    1;
    1;
    2;
    5,  1;
   14,  5,   4,  1;
   42, 21,  23, 14, 12,  5,  3;
  132, 84, 107, 82, 96, 55, 64, 37, 29, 22, 10, 0, 2;
  ...

Alois P. Heinz, Rows n = 0..10, flattened
Miklós Bóna, The Number of Permutations with Exactly r 132-Subsequences Is P-Recursive in the Size!, Advances in Applied Mathematics, Volume 18, Issue 4, May 1997, Pages 510-522.
FindStat - Combinatorial Statistic Finder, The number of occurrences of the pattern 312 in a permutation, The number of occurrences of the pattern 213 in a permutation, The number of occurrences of the pattern 231 in a permutation, The number of occurrences of the pattern 132 in a permutation
T. Mansour and A. Vainshtein, Counting occurrences of 132 in a permutation, arXiv:math/0105073 [math.CO], 2001.

Cf. A000108, A000142, A001810, A002054, A082970, A082971, A138162, A138159, A138163.

Join@@Array[Table[Length@Select[Permutations@Range@#,Length@Select[Subsets[#,{3}],Ordering@Ordering@#=={3,1,2}&]==k&],{k,0,Binomial[#+1,3]}]//.{a__,0}:>{a}&,8,0]  (* Giorgos Kalogeropoulos, Mar 26 2021 *)

Sum_{k>0} k * T(n,k) = A001810(n). - Alois P. Heinz, Oct 27 2015

More terms from Alois P. Heinz, Oct 26 2015

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).

Original entry on oeis.org

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).

A001089 Number of permutations of [n] containing exactly 2 increasing subsequences of length 3.

Original entry on oeis.org

0, 0, 0, 0, 3, 24, 133, 635, 2807, 11864, 48756, 196707, 783750, 3095708, 12152855, 47500635, 185082495, 719559600, 2793121080, 10830450780, 41965864794, 162539516448, 629399492330, 2437072038302, 9437097796918

Offset: 0

John Thomas Noonan [ noonan(AT)euclid.math.temple.edu ]

nonn

			For n=4, there are 4! = 24 permutations of 1234. The identity permutation 1234 has four increasing subsequences of length 3 (123, 124, 134, and 234), and the permutation 2314 has only one increasing subsequence of length 3 (234). Only the permutations 1243, 1324, and 2134 have exactly two increasing subsequences of length 3, and since there are three of them, a(4) = 3. - _Michael B. Porter_, Sep 03 2016

G. C. Greubel, Table of n, a(n) for n = 0..1000
M. Fulmek, Enumeration of permutations containing a prescribed number of occurrences of a pattern of length three, arXiv:math/0112092 [math.CO], 2001-2002. Also doi:10.1016/S0196-8858(02)00501-8, Adv. Appl. Math., 30, 2003, 607-632.
T. Mansour and A. Vainshtein, Counting occurrences of 123 in a permutation, arXiv:math/0105073 [math.CO], 2001.
Toufik Mansour, Sherry H. F. Yan and Laura L. M. Yang, Counting occurrences of 231 in an involution, Discrete Mathematics 306 (2006), pages 564-572.
J. Noonan and D. Zeilberger, The Enumeration of Permutations With a Prescribed Number of "Forbidden" Patterns, arXiv:math/9808080 [math.CO], 1998. Also doi:10.1006/aama.1996.0016, Adv. in Appl. Math. 17 (1996), no. 4, 381--407. MR1422065 (97j:05003).

Cf. A003517, A084249, A138159.

Leading column of A229158.

Concatenation([0,0,0,0], List([4..30], n-> (100+117*n+59*n^2)* Binomial(2*n,n-4)/(2*n*(2*n-1)*(n+5)))); # G. C. Greubel, Sep 19 2019

[0,0,0,0] cat [(100+117*n+59*n^2)*Binomial(2*n,n-4)/(2*n*(2*n-1)*(n+5)): n in [4..30]]; // G. C. Greubel, Sep 19 2019

seq(`if`(n=0, 0, (100+117*n+59*n^2)*binomial(2*n, n-4)/(2*n*(2*n-1)*(n+5))), n = 0..30); # G. C. Greubel, Sep 19 2019

{0}~Join~CoefficientList[Series[((x^5-3x^4+5x^3-10x^2+6*x-1)(1-4x)^(1/2) - 5x^5+7x^4-17x^3+20x^2-8*x+1)/(2x^6), {x,0,23}], x] (* or *)
{0}~Join~CoefficientList[Series[x^5*((1-(1-4x)^(1/2))/(2x))^11 +3x^3*( (1-(1-4x)^(1/2))/(2x))^8, {x,0,23}], x] (* Michael De Vlieger, Sep 03 2016 *)

a(n) = (100+117*n+59*n^2)*binomial(2*n,n-4)/(2*n*(2*n-1)*(n+5)) \\ G. C. Greubel, Sep 19 2019

[0,0,0,0]+[(100+117*n+59*n^2)*binomial(2*n,n-4)/(2*n*(2*n-1)*(n+5)) for n in (4..30)] # G. C. Greubel, Sep 19 2019

Noonan and Zeilberger conjectured that a(n) = ((59*n^2+117*n+100) /(2*n*(2*n-1)*(n+5))) *binomial(2*n,n-4). This was proved by Fulmek.
G.f.: ((x^5 -3*x^4 +5*x^3 -10*x^2 +6*x -1)*(1-4*x)^(1/2) - 5*x^5 +7*x^4 -17*x^3 +20*x^2 -8*x +1)/(2*x^6). - Mark van Hoeij, Oct 25 2011
G.f.: x^5*C(x)^11 + 3*x^3*C(x)^8, where C(x) is g.f. for the Catalan numbers (A000108). - Michael D. Weiner, Sep 02 2016
D-finite with recurrence -(n+5)*(n-4)*(59*n^2-n+42)*a(n) +2*(n-1)*(2*n-3)*(59*n^2 +117*n+100)*a(n-1) = 0, equivalent to recurrence of [Noonan-Zeilberger] binomial. - R. J. Mathar, Jan 04 2017

Terms a(25) onward added by G. C. Greubel, Sep 19 2019

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).

A342865 Irregular table read by rows: T(n,k) is the number of permutations in S_n that have exactly k occurrences of the pattern 1234. 0 <= k <= A000332(n).

Original entry on oeis.org

1, 1, 2, 6, 23, 1, 103, 12, 4, 0, 0, 1, 513, 102, 63, 10, 6, 12, 8, 0, 0, 5, 0, 0, 0, 0, 0, 1, 2761, 770, 665, 196, 146, 116, 142, 46, 10, 72, 32, 24, 0, 13, 0, 12, 18, 0, 0, 10, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 0

Peter Kagey, Mar 26 2021

Equivalently the table for the pattern 4321.

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,      12,       4,       0,       0,       1;
   6 |     513,     102,      63,      10,       6,      12,       8, ...
   7 |    2761,     770,     665,     196,     146,     116,     142, ...
   8 |   15767,    5545,    5982,    2477,    2148,    1204,    1782, ...
   9 |   94359,   39220,   49748,   25886,   25190,   13188,   19936, ...
  10 |  586590,  276144,  396642,  244233,  260505,  142550,  210663, ...
  11 | 3763290, 1948212, 3089010, 2167834, 2493489, 1476655, 2136586, ...

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. A000332, A005802.

Analogous for other patterns: A008302 (12), A138159 (321), A263771 (312), A342840 (1342), A342860 (2413), A342861 (1324), A342862 (2143), A342863 (1243), A342864 (1432).

cp's OEIS Frontend

A056986 Number of permutations on {1,...,n} containing any given pattern alpha in the symmetric group S_3.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

SageMath

Formula

A263771 Triangle read by rows: T(n,k) (n>=0, k>=0) is the number of permutations of n and k occurrences of the pattern 312.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

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).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A001089 Number of permutations of [n] containing exactly 2 increasing subsequences of length 3.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

Extensions

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).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

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).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

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).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

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).

Views

Author

Keywords