Len Smiley - cp's OEIS frontend

A111275 Number of inequivalent non-crossing partitions of n (equally spaced) points on a circle, under rotations and reflections.

1, 2, 3, 6, 10, 24, 49, 130, 336, 980, 2904, 9176, 29432, 97356, 326399, 1111770, 3825238, 13293456, 46553116, 164200028, 582706692, 2079517924, 7458493728, 26874412064, 97241528200, 353223728624, 1287668381250, 4709805627484

Offset: 1

David Callan and Len Smiley, Oct 21 2005

nonn

These may be viewed as bracelets (able to be turned over in space) designed with n beads on a circle, each of which is a vertex of exactly one of a set of non-touching internal polygons (which may be 1-gons (beads), 2-gons (2 connected beads), etc.).

S.-C. Chang, J. L. Jacobsen, J. Salas, R. Shrock, "Exact Potts model partition functions for strips of the triangular lattice", J. Statist. Phys. 114, nos.3-4, pp. 763-823 [Corollary 2.1]
Motzkin, T. "Relations Between Hypersurface Cross Ratios and a Combinatorial Formula for Partitions of a Polygon for Permanent Preponderance and for Non-Associative Products." Bull. Amer. Math. Soc. 54, page 360, 1948.

D. Callan and L. Smiley, Non-crossing Partitions under Rotation and Reflection
Tilman Piesk, Partition related number triangles
L. Smiley, a(6)
L. Smiley, a(5)

Cf. A209612.

Table[Length[EquivalenceClasses[NCPartitions[n], groupDihedral[n]]], {n, 9}]

(A054357(n) + A001405(n))/2.

A111282 Number of permutations avoiding the patterns {1432,2431,3412,3421,4132,4231,4312,4321}; number of strong sorting class based on 1432.

Original entry on oeis.org

1, 2, 6, 16, 42, 110, 288, 754, 1974, 5168, 13530, 35422, 92736, 242786, 635622, 1664080, 4356618, 11405774, 29860704, 78176338, 204668310, 535828592, 1402817466, 3672623806, 9615053952, 25172538050, 65902560198, 172535142544

Offset: 1

Len Smiley, Nov 01 2005

a(n-1) is the sum, over all Boolean n-strings, of the product of the lengths of the runs. For example, the Boolean 7-string (0,1,1,0,1,1,1) has four runs, whose lengths are 1,2,1 and 3, contributing a product of 6 to a(6). The 4 Boolean 2-strings contribute to a(3) as follows: 00 and 11 both contribute 2 and 01 and 10 both contribute 1. - David Callan, Jul 22 2008
a(n) = A025169(n-2) for n > 1. - Reinhard Zumkeller, Apr 08 2012
The sequence 0, 2, 0, 0, 1, 2, 6, 16, 42, 110, 288, 754, 1974, ... with g.f. H(x) = 2*x+(x^4-x^5+x^6)/(1-3*x+x^2) is the number of "splitted indecomposable weakly threshold graphs" on n nodes [Barrus, 2016]. - N. J. A. Sloane, Jul 25 2017
Number of permutations of length n>0 avoiding the partially ordered pattern (POP) {2>1, 2>4} of length 4. That is, number of length n permutations having no subsequences of length 4 in which the second element is larger than the first and fourth elements. - Sergey Kitaev, Dec 09 2020

			x + 2*x^2 + 6*x^3 + 16*x^4 + 42*x^5 + 110*x^6 + 288*x^7 + ...

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
M. Albert, R. Aldred, M. Atkinson, C. Handley, D. Holton, D. McCaughan and H. van Ditmarsch, Sorting Classes, Elec. J. of Comb. 12 (2005) #R31.
Michael D. Barrus, Weakly threshold graphs, arXiv preprint arXiv:1608.01358 [math.CO], 2016.
Hacène Belbachir, Soumeya Merwa Tebtoub, and László Németh, Ellipse Chains and Associated Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.
Alice L. L. Gao, Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, arXiv:1903.08946 [math.CO], 2019.
Alice L. L. Gao, Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, The Electronic Journal of Combinatorics 26(3) (2019), P3.26.
Index entries for linear recurrences with constant coefficients, signature (3, -1).

a111282 n = a111282_list !! (n-1)
a111282_list = 1 : a025169_list
-- Reinhard Zumkeller, Apr 08 2012

a[1] = 1; a[2] = 2; a[3] = 6; a[n_] := a[n] = 3a[n - 1] - a[n - 2]; Table[a[n], {n, 28}] (* Robert G. Wilson v *)

a(n) = 3a(n-1) - a(n-2), n > 3.
a(n) = A025169(n-2), n > 1. - R. J. Mathar, Aug 18 2008
From Paul Barry, Oct 13 2009: (Start)
G.f.: (1 - x + x^2)/(1 - 3x + x^2).
a(n) = F(2n+1) + F(2n-2) + 0^n. (End)

A111281 Number of permutations avoiding the patterns {2413,2431,4213,3412,3421,4231,4321,4312}; number of strong sorting class based on 2413.

Original entry on oeis.org

1, 1, 2, 6, 16, 40, 100, 252, 636, 1604, 4044, 10196, 25708, 64820, 163436, 412084, 1039020, 2619764, 6605420, 16654772, 41993004, 105880308, 266964460, 673118772, 1697188012, 4279255412, 10789627756, 27204748468, 68593500716, 172950260724, 436073277676

Offset: 0

Len Smiley, Nov 01 2005

a(n) = term (1,1) in M^n, M = the 4x4 matrix [1,1,1,1; 0,1,0,1; 0,0,1,1; 1,0,0,1]. - Gary W. Adamson, Apr 29 2009

Number of permutations of length n>0 avoiding the partially ordered pattern (POP) {1>2, 1>4} of length 4. That is, number of length n permutations having no subsequences of length 4 in which the first element is larger than the second and fourth elements. - Sergey Kitaev, Dec 09 2020

Michael De Vlieger, Table of n, a(n) for n = 0..2490
M. Albert, R. Aldred, M. Atkinson, C Handley, D. Holton, D. McCaughan and H. van Ditmarsch, Sorting Classes, Elec. J. of Comb. 12 (2005) R31.
Alice L. L. Gao, Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, arXiv:1903.08946 [math.CO], 2019.
Alice L. L. Gao, Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, The Electronic Journal of Combinatorics 26(3) (2019), P3.26.
Kai Ting Keshia Yap, David Wehlau, and Imed Zaguia, Permutations Avoiding Certain Partially-ordered Patterns, arXiv:2101.12061 [math.CO], 2021.
Index entries for linear recurrences with constant coefficients, signature (3,-2,2).

a[1] = 1; a[2] = 2; a[3] = 6; a[n_] := a[n] = 3a[n - 1] - 2a[n - 2] + 2a[n - 3]; Table[a[n], {n, 28}] (* Robert G. Wilson v *)

a(n) = 3*a(n-1)-2*a(n-2)+2*a(n-3).

G.f.: 1+x*(1-x+2*x^2)/(1-3*x+2*x^2-2*x^3). - Colin Barker, Jan 16 2012

More terms from Robert G. Wilson v, Nov 04 2005

a(0)=1 prepended by Alois P. Heinz, May 07 2021

A111279 Number of permutations avoiding the patterns {3241,3421,4321}; number of weak sorting class based on 3241.

Original entry on oeis.org

1, 1, 2, 6, 21, 79, 309, 1237, 5026, 20626, 85242, 354080, 1476368, 6173634, 25873744, 108628550, 456710589, 1922354351, 8098984433, 34147706833, 144068881455, 608151037123, 2568318694867, 10850577045131, 45856273670841, 193850277807569, 819669810565949

Offset: 0

Len Smiley, Nov 01 2005

nonn

Is this the same sequence as A026737? - Andrew S. Plewe, May 09 2007
Yes, see the Callan reference "A bijection...". - Joerg Arndt, Feb 29 2016
a(n) is the number of permutations of length n avoiding the partially ordered pattern (POP) {1>3, 1>4, 3>2} of length 4. That is, the number of length n permutations having no subsequences of length 4 in which the first element is the largest and the third element is larger than the second element. - Sergey Kitaev, Dec 10 2020

			a(4) = 21 since the top row terms of M^3 = (11, 6, 3, 1, 0, 0, 0, ...)

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (terms n=1..300 from Vincenzo Librandi)
M. Albert, R. Aldred, M. Atkinson, C. Handley, D. Holton, D. McCaughan and H. van Ditmarsch, Sorting Classes, Elec. J. of Comb. 12 (2005) R31.
David Callan, A bijection for two sequences in OEIS, arXiv:1602.08347 [math.CO], 2016.
David Callan, and Toufik Mansour, Five subsets of permutations enumerated as weak sorting permutations, arXiv:1602.05182 [math.CO], 2016.
Alice L. L. Gao and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, arXiv:1903.08946 [math.CO], 2019.
Alice L. L. Gao, and Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, The Electronic Journal of Combinatorics 26(3) (2019), P3.26.

Rest[ CoefficientList[ Series[(3 - 13x + 2x^2 + (5x - 1)*Sqrt[1 - 4x])/(2*(1 - 4x - x^2)), {x, 0, 24}], x]] (* Robert G. Wilson v, Nov 04 2005 *)

O.g.f.: (3-13*x+2*x^2+(5*x-1)*sqrt(1-4*x))/(2*(1-4*x-x^2)).
From Gary W. Adamson, Nov 14 2011: (Start)
a(n) is the sum of top row terms of M^(n-1), M is an infinite square production matrix with powers of 2 as the left border as follows:
  1, 1, 0, 0, 0, ...
  2, 1, 1, 0, 0, ...
  4, 1, 1, 1, 0, ...
  8, 1, 1, 1, 1, ...
  ... (End)
The top rows of these matrix powers, 1; 1,1; 3,2,1; 11,6,3,1; 43,21,10,4,1; appear also as columns in A026736. - R. J. Mathar, Nov 15 2011
D-finite with recurrence n*a(n) + (16-13*n)*a(n-1)+(55*n-134)*a(n-2) + (264-71*n)*a(n-3) + 10*(7-2*n)*a(n-4) = 0. - R. J. Mathar, Nov 15 2011
Shorter recurrence: n*(n+5)*a(n) = 2*(4*n^2 + 17*n - 30)*a(n-1) - 3*(5*n^2 + 17*n - 80)*a(n-2) - 2*(n+6)*(2*n-5)*a(n-3). - Vaclav Kotesovec, Oct 18 2012
a(n) ~ (5/2-11/10*sqrt(5))*(sqrt(5)+2)^n. - Vaclav Kotesovec, Oct 18 2012

More terms from Robert G. Wilson v, Nov 04 2005

a(0)=1 prepended by Alois P. Heinz, Dec 11 2020

A111284 Number of permutations of [n] avoiding the patterns {2143, 2341, 2413, 2431, 3142, 3241, 3412, 3421, 4123, 4213, 4231, 4321, 4132, 4312}; number of strong sorting classes based on 2143.

Original entry on oeis.org

1, 2, 6, 10, 14, 18, 22, 26, 30, 34, 38, 42, 46, 50, 54, 58, 62, 66, 70, 74, 78, 82, 86, 90, 94, 98, 102, 106, 110, 114, 118, 122, 126, 130, 134, 138, 142, 146, 150, 154, 158, 162, 166, 170, 174, 178, 182, 186, 190, 194, 198, 202, 206, 210, 214, 218, 222, 226, 230

Offset: 1

Len Smiley, Nov 01 2005

This sequence might also be called "The Non-Pythagorean integers" since no primitive Pythagorean triangle (PPT) exists containing them. Numbers of the form 4n-2 cannot be a leg or hypotenuse of PPT [a,b,c]. This excludes all even members of the present sequence. Integers 1 and zero are excluded because they form a 'degenerate triangle' with angles = 0. Compare A125667. - H. Lee Price, Feb 02 2007

Besides the first term this sequence is the denominator of Pi/8 = 1/2 - 1/6 + 1/10 - 1/14 + 1/18 - 1/22 + .... - Mohammad K. Azarian, Oct 14 2011

Mohammad K. Azarian, Problem 1218, Pi Mu Epsilon Journal, Vol. 13, No. 2, Spring 2010, p. 116. Solution published in Vol. 13, No. 3, Fall 2010, pp. 183-185.
Granino A. Korn and Theresa M. Korn, Mathematical Handbook for Scientists and Engineers, McGraw-Hill Book Company, New York (1968).

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
M. Albert, R. Aldred, M. Atkinson, C Handley, D. Holton, D. McCaughan and H. van Ditmarsch, Sorting Classes, Elec. J. of Comb. 12 (2005) R31
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A125667. Complement of the union of {1}, A020882, A020883 and A020884.

Cf. A016825, A161718.

Table[If[n == 1, 1, 4n - 6], {n, 60}] (* Robert G. Wilson v, Nov 04 2005 *)

a(n) = 4*n-6, n>=2.
a(n) = A016825(n-2), n>1. - R. J. Mathar, Aug 18 2008
G.f.: x(1+3x^2)/(1-x)^2. - R. J. Mathar, Nov 10 2008
a(n^2 - 2n + 3)/2 = Sum_{i=1..n} a(i). - Ivan N. Ianakiev, Apr 24 2013
a(n) = 2*a(n-1) - a(n-2), n>3. - Rick L. Shepherd, Jul 06 2017
a(n) = |A161718(n-1)| = (-1)^(n-1)*A161718(n-1), n>0. - Rick L. Shepherd, Jul 06 2017
E.g.f.: 3*(x + 2) + exp(x)*(4*x - 6). - Stefano Spezia, Feb 02 2023

A111285 Number of permutations avoiding the patterns {2431, 3421, 4231, 4321, 24513, 42513, 34512, 43512}; number of strong sorting class based on 2431.

Original entry on oeis.org

1, 1, 2, 6, 20, 66, 216, 706, 2308, 7546, 24672, 80666, 263740, 862306, 2819336, 9217906, 30138228, 98537866, 322172592, 1053353226, 3443970860, 11260168946, 36815469656, 120369313506, 393551182948, 1286727730586, 4206996000512

Offset: 0

Len Smiley, Nov 01 2005

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
M. Albert, R. Aldred, M. Atkinson, C Handley, D. Holton, D. McCaughan and H. van Ditmarsch, Sorting Classes, Elec. J. of Comb., Vol. 12 (2005), R31.
Index entries for linear recurrences with constant coefficients, signature (4,-3,2).

I:=[1, 2, 6]; [n le 3 select I[n] else 4*Self(n-1)-3*Self(n-2)+2*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jun 27 2012

a[1] = 1; a[2] = 2; a[3] = 6; a[n_] := a[n] = 4a[n - 1] - 3a[n - 2] + 2a[n - 3]; Table[a[n], {n, 26}] (* Robert G. Wilson v *)
CoefficientList[Series[(1-2*x+x^2)/(1-4*x+3*x^2-2*x^3),{x,0,40}],x] (* or *) LinearRecurrence[{4,-3,2},{1,2,6},40] (* Vincenzo Librandi, Jun 27 2012 *)

a(n) = 4*a(n-1) - 3*a(n-2) + 2*a(n-3), n>=4.
G.f.: 1+x*(1-x)^2/(1-4*x+3*x^2-2*x^3).
a(n) = A175005(n)+A175005(n-2)-2*A175005(n-1). - R. J. Mathar, Aug 19 2022~

a(0)=1 prepended by Alois P. Heinz, Mar 12 2024

A111280 Number of permutations avoiding the patterns {4231, 4321, 35142, 45312, 42513, 45132, 35412, 45213, 43512, 456123, 351624, 451623, 356124}; number of strong sorting class based on 4231.

Original entry on oeis.org

1, 1, 2, 6, 22, 113, 431, 1584, 5920, 22214, 83239, 311777, 1167902, 4375090, 16389450, 61395989, 229993639, 861572476, 3227511492, 12090486122, 45291815419, 169666341761, 635582108218, 2380935499534, 8919152662622, 33411776268873

Offset: 0

Len Smiley, Nov 01 2005

M. Albert, R. Aldred, M. Atkinson, C Handley, D. Holton, D. McCaughan and H. van Ditmarsch, Sorting Classes, Elec. J. of Comb., Vol. 12 (2005), R31.
Index entries for linear recurrences with constant coefficients, signature (4,-2,4,0,-1).

a[1] = 1; a[2] = 2; a[3] = 6; a[4] = 22; a[5] = 113; a[n_] := a[n] = 4a[n - 1] - 2a[n - 2] + 4a[n - 3] - a[n - 5]; Table[ a[n], {n, 25}] (* Robert G. Wilson v, Nov 04 2005 *)
LinearRecurrence[{4,-2,4,0,-1},{1,2,6,22,113},30] (* Harvey P. Dale, Jun 03 2019 *)

a(n) = 4*a(n-1)-2*a(n-2)+4*a(n-3)-a(n-5).

G.f.: 1+x*(1-2*x-2*x^3+29*x^4)/(1-4*x+2*x^2-4*x^3+x^5). - Colin Barker, Jan 16 2012

More terms from Robert G. Wilson v, Nov 04 2005

a(0)=1 prepended by Alois P. Heinz, Mar 12 2024

A111286 Number of permutations avoiding the patterns {1342, 1432, 2341, 2431, 3142, 3241, 3412, 3421, 4132, 4231, 4312, 4321}; number of strong sorting class based on 1342.

Original entry on oeis.org

1, 1, 2, 6, 12, 24, 48, 96, 192, 384, 768, 1536, 3072, 6144, 12288, 24576, 49152, 98304, 196608, 393216, 786432, 1572864, 3145728, 6291456, 12582912, 25165824, 50331648, 100663296, 201326592, 402653184, 805306368, 1610612736, 3221225472

Offset: 0

Len Smiley, Nov 01 2005

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
M. Albert, R. Aldred, M. Atkinson, C Handley, D. Holton, D. McCaughan and H. van Ditmarsch, Sorting Classes, Elec. J. of Comb., Vol. 12 (2005), R31.
Index entries for linear recurrences with constant coefficients, signature (2).

Cf. A003945, A007283, A042950, A098011, A110164 - differs from each by one initial term.

Table[If[n == 1, 1, If[n == 2, 2, 3*2^(n - 2)]], {n, 32}] (* Robert G. Wilson v *)
LinearRecurrence[{2},{1,2,6},40] (* Harvey P. Dale, Jul 14 2019 *)

a(n) = 3*2^(n-2), n>=3.

a(n) = 2*a(n-1) for n=3. G.f.: (1-x+2*x^3)/(1-2*x). - Colin Barker, Nov 29 2012

a(0)=1 prepended by Alois P. Heinz, Mar 12 2024

A111276 Number of chiral non-crossing partition patterns of n points on a circle, divided by 2.

Original entry on oeis.org

0, 0, 0, 0, 0, 4, 14, 60, 210, 728, 2442, 8252, 27716, 93924, 319964, 1098900, 3800928, 13244836, 46460738, 164015272, 582353976, 2078812492, 7457141650, 26871707908, 97236327900, 353213328024, 1287648322950, 4709765510884, 17279999438748, 63583033400968

Offset: 1

David Callan and Len Smiley, Oct 21 2005

nonn

Half of the number of those rotation-inequivalent patterns of non-crossing partitions of n (equally spaced) points on a circle which are not invariant under reflections. Division by two counts one pattern from each chiral (Right-handed,Left-handed) pair.

Andrew Howroyd, Table of n, a(n) for n = 1..1000
D. Callan and L. Smiley, Non-crossing Partitions under Rotation and Reflection, arXiv:math/0510447 [math.CO], 2005.
L. Smiley, a(5) = 0
L. Smiley, a(6)=8/2=4

Cf. A001405, A054357, A111275.

a[n_] := If[n < 6, 0, ((Binomial[2n, n]/(n+1) + DivisorSum[n, Binomial[2#, #] EulerPhi[n/#] Boole[# < n]&])/n - Binomial[n, Floor[n/2]])/2];
Array[a, 22] (* Jean-François Alcover, Feb 17 2019 *)

a(n) = (sumdiv(n, d, eulerphi(n/d)*binomial(2*d, d))/n - binomial(2*n, n)/(n+1) - binomial(n,n\2))/2 \\ Andrew Howroyd, Nov 19 2024

a(n) = (A054357(n) - A001405(n))/2.

a(23) onwards from Andrew Howroyd, Nov 19 2024

A111283 Number of permutations avoiding the patterns {4321, 45132, 45231, 35412, 53412, 45213, 43512, 45312, 456123, 451623, 356124}; number of strong sorting class based on 4321.

Original entry on oeis.org

1, 1, 2, 6, 23, 96, 409, 1751, 7505, 32176, 137956, 591501, 2536132, 10873981, 46623553, 199904321, 857114814, 3674987126, 15756967635, 67559972476, 289671844661, 1242004318751, 5325249092137, 22832672531956, 97897943538708

Offset: 0

Len Smiley, Nov 01 2005

nonn

M. Albert, R. Aldred, M. Atkinson, C Handley, D. Holton, D. McCaughan and H. van Ditmarsch, Sorting Classes, Elec. J. of Comb., Vol. 12 (2005), R31.
Index entries for linear recurrences with constant coefficients, signature (5,-3,0,-1).

a[1] = 1; a[2] = 2; a[3] = 6; a[n_] := a[n] = 4a[n - 1] + a[n - 2] + a[n - 3] - 4; Table[a[n], {n, 24}] (* Robert G. Wilson v, Nov 04 2005 *)
LinearRecurrence[{5,-3,0,-1},{1,2,6,23},30] (* Harvey P. Dale, Jan 01 2017 *)

a(n) = 4*a(n-1) + a(n-2) + a(n-3) - 4; n>=4.
G.f.: 1+x*(1-3*x-x^2-x^3)/((1-x)*(1-4*x-x^2-x^3)). - Colin Barker, Jan 16 2012
a(n) = 5*a(n-1) - 3*a(n-2) - a(n-4). - Wesley Ivan Hurt, Aug 04 2025

More terms from Robert G. Wilson v, Nov 04 2005

a(0)=1 prepended by Alois P. Heinz, Mar 12 2024

cp's OEIS Frontend

User: Len Smiley

A111275 Number of inequivalent non-crossing partitions of n (equally spaced) points on a circle, under rotations and reflections.

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A111282 Number of permutations avoiding the patterns {1432,2431,3412,3421,4132,4231,4312,4321}; number of strong sorting class based on 1432.

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A111281 Number of permutations avoiding the patterns {2413,2431,4213,3412,3421,4231,4321,4312}; number of strong sorting class based on 2413.

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A111279 Number of permutations avoiding the patterns {3241,3421,4321}; number of weak sorting class based on 3241.

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A111284 Number of permutations of [n] avoiding the patterns {2143, 2341, 2413, 2431, 3142, 3241, 3412, 3421, 4123, 4213, 4231, 4321, 4132, 4312}; number of strong sorting classes based on 2143.

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A111285 Number of permutations avoiding the patterns {2431, 3421, 4231, 4321, 24513, 42513, 34512, 43512}; number of strong sorting class based on 2431.

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A111280 Number of permutations avoiding the patterns {4231, 4321, 35142, 45312, 42513, 45132, 35412, 45213, 43512, 456123, 351624, 451623, 356124}; number of strong sorting class based on 4231.

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A111286 Number of permutations avoiding the patterns {1342, 1432, 2341, 2431, 3142, 3241, 3412, 3421, 4132, 4231, 4312, 4321}; number of strong sorting class based on 1342.

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