A264319 -id:A264319 - cp's OEIS frontend

A264173 Number T(n,k) of permutations of [n] with exactly k (possibly overlapping) occurrences of the consecutive pattern 1324; triangle T(n,k), n>=0, 0<=k<=max(0,floor(n/2-1)), read by rows.

1, 1, 2, 6, 23, 1, 110, 10, 632, 86, 2, 4229, 782, 29, 32337, 7571, 407, 5, 278204, 78726, 5856, 94, 2659223, 882997, 84351, 2215, 14, 27959880, 10657118, 1251246, 48234, 322, 320706444, 137977980, 19318314, 984498, 14322, 42, 3985116699, 1910131680, 311306106

Offset: 0

Alois P. Heinz, Nov 06 2015

Pattern 4231 gives the same triangle.

			T(4,1) = 1: 1324.
T(6,2) = 2: 132546, 142536.
T(8,3) = 5: 13254768, 13264758, 14253768, 14263758, 15263748.
T(10,4) = 14: 132547698(10), 132548697(10), 132647598(10), 132648597(10), 132748596(10), 142537698(10), 142538697(10), 142637598(10), 142638597(10), 142738596(10), 152637498(10), 152638497(10), 152738496(10), 162738495(10).
Triangle T(n,k) begins:
00 :        1;
01 :        1;
02 :        2;
03 :        6;
04 :       23,      1;
05 :      110,     10;
06 :      632,     86,     2;
07 :     4229,    782,    29;
08 :    32337,   7571,   407,    5;
09 :   278204,  78726,  5856,   94;
10 :  2659223, 882997, 84351, 2215, 14;

Alois P. Heinz, Rows n = 0..120, flattened

Columns k=0-10 give: A113228, A264174, A264175, A264176, A264177, A264178, A264179, A264180, A264181, A264182, A264183.
Row sums give A000142.
T(2n+2,n) gives A000108(n) for n>0.
Cf. A004526, A061206, A264319 (pattern 3412).

b:= proc(u, o, t) option remember; expand(`if`(u+o=0, 1,
      add(b(u-j, o+j-1, `if`(t>0 and j (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0$2)):
seq(T(n), n=0..14);

b[u_, o_, t_] := b[u, o, t] = Expand[If[u + o == 0, 1, Sum[b[u - j, o + j - 1, If[t > 0 && j < t, -j, 0]], {j, 1, u}] + Sum[b[u + j - 1, o - j, j] * If[t < 0 && -j <= t, x, 1], {j, 1, o}]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][ b[n, 0, 0]];
Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Apr 30 2017, translated from Maple *)

Sum_{k>0} k * T(n,k) = ceiling((n-3)*n!/4!) = A061206(n-3) (for n>3).

A113229 Number of permutations avoiding the consecutive pattern 3412.

Original entry on oeis.org

1, 1, 2, 6, 23, 110, 631, 4223, 32301, 277962, 2657797, 27954521, 320752991, 3987045780, 53372351265, 765499019221, 11711207065229, 190365226548070, 3276401870322033, 59523410471007913, 1138295039078030599, 22856576346825690128, 480807130959249565541

Offset: 0

David Callan, Oct 19 2005

nonn

a(n) is the number of permutations on [n] that avoid the consecutive pattern 3412 (also number that avoid 2143).

			The 5! - a(5) = 10 permutations on [5] not counted by a(5) are 14523, 24513, 34125, 34512, 35124, 43512, 45123, 45132, 45231, 53412.

Alois P. Heinz, Table of n, a(n) for n = 0..200 (terms n = 0..60 from Ray Chandler)
A. Baxter, B. Nakamura, and D. Zeilberger. Automatic generation of theorems and proofs on enumerating consecutive Wilf-classes
V. Dotsenko and A. Khoroshkin, Shuffle algebras, homology, and consecutive pattern avoidance, arXiv preprint arXiv:1109.2690 [math.CO], 2011.
Sergi Elizalde, Asymptotic enumeration of permutations avoiding generalized patterns, arXiv:math/0505254 [math.CO], 2005.
Sergi Elizalde, Asymptotic enumeration of permutations avoiding generalized patterns, Adv. in Appl. Math. 36 (2006), no. 2, 138-155.
S. Elizalde and M. Noy, Consecutive patterns in permutations, Adv. Appl. Math. 30 (2003), 110-125.
Steven Finch, Pattern-Avoiding Permutations [Broken link?]
Steven Finch, Pattern-Avoiding Permutations [Cached copy, with permission]

Cf. A113228, A117156, A117158, A117226, A201692, A201693.

Column k=0 of A264319.

b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
      add(b(u-j, o+j-1, `if`(t>0 and j>t, t-j, 0)), j=1..u)+
      add(b(u+j-1, o-j, j), j=`if`(t<0,1-t,1)..o))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..25);  # Alois P. Heinz, Nov 07 2013

b[u_, o_, t_] := b[u, o, t] = If[u+o == 0, 1, Sum[b[u-j, o+j-1, If[t>0 && j>t, t-j, 0]], {j, 1, u}] + Sum[b[u+j-1, o-j, j], {j, Range[If[t<0, 1-t, 1], o]}]]; a[n_] := b[n, 0, 0]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Mar 13 2015, after Alois P. Heinz *)

The Dotsenko et al. reference gives a g.f. There is an associated triangle of numbers c_{n,l} that should be added to the OEIS if it is not already present.

a(n) ~ c * d^n * n!, where d = 0.9561742431150784273897350385923872770208469..., c = 1.1465405299007850875068632404058971045769... . - Vaclav Kotesovec, Aug 23 2014

A264320 Number of permutations of [n] with exactly one occurrence of the consecutive pattern 3412.

Original entry on oeis.org

1, 10, 88, 794, 7639, 79164, 885128, 10657588, 137752283, 1904555934, 28068929900, 439495830750, 7288494061255, 127654030808304, 2355066281798588, 45655570757683888, 927984709909530539, 19735811069218548106, 438344811235148801632, 10149979837759608190794

Offset: 4

Alois P. Heinz, Nov 11 2015

nonn

			a(4) = 1: 3412.
a(5) = 10: 14523, 24513, 34125, 34512, 35124, 43512, 45123, 45132, 45231, 53412.

Alois P. Heinz, Table of n, a(n) for n = 4..200

Column k=1 of A264319.

A264321 Number of permutations of [n] with exactly two (possibly overlapping) occurrences of the consecutive pattern 3412.

Original entry on oeis.org

1, 23, 379, 5706, 84354, 1266150, 19621124, 316459848, 5331647951, 93959680785, 1732274350449, 33392773007922, 672442070185276, 14130089667648160, 309458726679280550, 7054945428667834408, 167218369471952249237, 4115763361988379577315, 105071634182596234984655

Offset: 6

Alois P. Heinz, Nov 11 2015

nonn

			a(6) = 1: 563412.
a(7) = 23: 1674523, 2674513, 3674512, 4673512, 5614723, 5624713, 5634127, 5634712, 5673412, 5714623, 5724613, 5734126, 5734612, 6573412, 6714523, 6724513, 6734125, 6734512, 6735124, 6745123, 6745132, 6745231, 7563412.

Alois P. Heinz, Table of n, a(n) for n = 6..200

Column k=2 of A264319.

A264322 Number of permutations of [n] with exactly three (possibly overlapping) occurrences of the consecutive pattern 3412.

Original entry on oeis.org

1, 48, 1520, 38452, 869740, 18720732, 396975296, 8458711900, 183197270195, 4060222577980, 92457026432364, 2168229942019192, 52433345808131318, 1308330112118893072, 33691149632085370112, 895280698879996177240, 24542614402374373350957, 693775997089974201665016

Offset: 8

Alois P. Heinz, Nov 11 2015

nonn

			a(8) = 1: 78563412.
a(9) = 48: 189674523, 289674513, 389674512, ..., 896745132, 896745231, 978563412.

Alois P. Heinz, Table of n, a(n) for n = 8..180

Column k=3 of A264319.

A264323 Number of permutations of [n] with exactly four (possibly overlapping) occurrences of the consecutive pattern 3412.

Original entry on oeis.org

1, 89, 5461, 238356, 8368831, 259770975, 7542478135, 211794493290, 5870838954710, 162745118970610, 4549825111681964, 128994813976541176, 3722621639161251074, 109622707753571938234, 3299453321046315967098, 101610430484526744949716, 3203896974449174366163207

Offset: 10

Alois P. Heinz, Nov 11 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 10..180

Column k=4 of A264319.

A264324 Number of permutations of [n] with exactly five (possibly overlapping) occurrences of the consecutive pattern 3412.

Original entry on oeis.org

1, 150, 17956, 1354134, 74605701, 3373438200, 134999600212, 5013535378432, 178086394182356, 6171760208699812, 211481116147026848, 7230935446550288908, 248287682074519332846, 8600459778995519305416, 301511114563838537243472, 10722994509016940787519096

Offset: 12

Alois P. Heinz, Nov 11 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 12..180

Column k=5 of A264319.

A264325 Number of permutations of [n] with exactly six (possibly overlapping) occurrences of the consecutive pattern 3412.

Original entry on oeis.org

1, 235, 57035, 7221886, 618249852, 40878791512, 2270356257626, 112176122124184, 5127825571576298, 222748847168538286, 9366990120105399606, 386330492305650221364, 15774418153169702657988, 642000537768645881284284, 26174608034919642252229176

Offset: 14

Alois P. Heinz, Nov 11 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 14..180

Column k=6 of A264319.

A264326 Number of permutations of [n] with exactly seven (possibly overlapping) occurrences of the consecutive pattern 3412.

Original entry on oeis.org

1, 348, 184964, 37331800, 4838911370, 464380098592, 35861849944448, 2368950978884920, 140048359125251970, 7656366194099685024, 396295578652586265936, 19754556228617612401848, 960237883152840490862664, 45936904818881537730689976, 2177814852070869127205627616

Offset: 16

Alois P. Heinz, Nov 11 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 16..180

Column k=7 of A264319.

A264327 Number of permutations of [n] with exactly eight (possibly overlapping) occurrences of the consecutive pattern 3412.

Original entry on oeis.org

1, 493, 633677, 191009848, 36439591193, 5004015747625, 534739648790073, 47271093160982634, 3626506774076237883, 250491072534377023959, 16015467370198042469229, 967671571943986351197684, 56117598468254134348216611, 3160386391272010574690621055

Offset: 18

Alois P. Heinz, Nov 11 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 18..180

Column k=8 of A264319.

cp's OEIS Frontend

A264173 Number T(n,k) of permutations of [n] with exactly k (possibly overlapping) occurrences of the consecutive pattern 1324; triangle T(n,k), n>=0, 0<=k<=max(0,floor(n/2-1)), read by rows.

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A113229 Number of permutations avoiding the consecutive pattern 3412.

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A264320 Number of permutations of [n] with exactly one occurrence of the consecutive pattern 3412.

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A264321 Number of permutations of [n] with exactly two (possibly overlapping) occurrences of the consecutive pattern 3412.

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A264322 Number of permutations of [n] with exactly three (possibly overlapping) occurrences of the consecutive pattern 3412.

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A264323 Number of permutations of [n] with exactly four (possibly overlapping) occurrences of the consecutive pattern 3412.

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A264324 Number of permutations of [n] with exactly five (possibly overlapping) occurrences of the consecutive pattern 3412.

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A264325 Number of permutations of [n] with exactly six (possibly overlapping) occurrences of the consecutive pattern 3412.

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A264326 Number of permutations of [n] with exactly seven (possibly overlapping) occurrences of the consecutive pattern 3412.

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A264327 Number of permutations of [n] with exactly eight (possibly overlapping) occurrences of the consecutive pattern 3412.

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