Vincent Pilaud - cp's OEIS frontend

A263791 Number of permutations of [n] avoiding the generalized patterns 1(k+2)-(u_1+1)-...-(u_k+1) for all permutations u of [k].

1, 1, 1, 1, 2, 1, 1, 2, 5, 1, 1, 2, 6, 14, 1, 1, 2, 6, 22, 42, 1, 1, 2, 6, 24, 92, 132, 1, 1, 2, 6, 24, 114, 420, 429, 1, 1, 2, 6, 24, 120, 612, 2042, 1430, 1, 1, 2, 6, 24, 120, 696, 3600, 10404, 4862, 1, 1, 2, 6, 24, 120, 720, 4512, 22680, 54954, 16796, 1, 1, 2, 6, 24, 120, 720, 4920, 31920, 150732, 298648, 58786, 1, 1, 2, 6, 24, 120, 720, 5040, 37200, 242160, 1045440

Offset: 1

Vincent Pilaud, Oct 26 2015

The sequence reads the antidiagonals of the table [a(n,k)] (for k >= 0 and n >= 1). See examples below.
a(n,k) is the number of permutations of [n] avoiding the generalized patterns 1(k+2)-(u_1+1)-...-(u_k+1) for all permutations u of [k].
a(n,k) is the number of classes of the k-twist congruence on S_n, defined as the transitive closure of the rewriting rule UacV_1b_1...V_kb_kW = UcaV_1b_1...V_kb_kW for a < b_1, ..., b_k < c in [n] and U, V_1, ..., V_k, W some (possibly empty) words on [n].
a(n,k) is the number of (k,n)-twists whose contact graph is acyclic. A (k,n)-twist is a reduced pipe dream for the permutation (1, ..., k, n+k, ..., k+1, n+k+1, ..., n+2k). The contact graph of a (k,n)-twist is the graph with a node for each pipe and an oriented arc for each elbow from the pipe passing southeast of the elbow to the pipe passing northwest of the elbow.
a(n,k) is the number of vertices of the brick polytope for the word c^k w_o(c) where c = 1 2 ... n-1 is the linear Coxeter element in type A.

			Table a(n,k) begins (row index n >= 1, column index k >= 0):
1     1      1       1       1       1       1       1       1       1 ...
1     2      2       2       2       2       2       2       2       2 ...
1     5      6       6       6       6       6       6       6       6 ...
1    14     22      24      24      24      24      24      24      24 ...
1    42     92     114     120     120     120     120     120     120 ...
1   132    420     612     696     720     720     720     720     720 ...
1   429   2042    3600    4512    4920    5040    5040    5040    5040 ...
1  1430  10404   22680   31920   37200   39600   40320   40320   40320 ...
1  4862  54954  150732  242160  305280  341280  357840  362880  362880 ...
1 16796 298648 1045440 1942800 2680800 3175200 3457440 3588480 3628800 ...
..........................................................................

V. Pilaud, Brick polytopes, lattice quotients, and Hopf algebras, arXiv:1505.07665 [math.CO], 2015.

Cf. A000108, A000142.

a(n,0) = 1.
a(n,1) = binomial(2n,n)/(n+1) (Catalan number A000108).
When n <= k+1, a(n,k) = n! (factorial A000142).

A180501 Triangle read by row. T(n,m) gives the number of isomorphism classes of arrangements of n pseudolines and m double pseudolines in the projective plane.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 9, 46, 2, 16, 265, 6998, 153528

Offset: 0

Vincent Pilaud, Sep 08 2010

J. Ferté, V. Pilaud and M. Pocchiola, On the number of arrangements of five double pseudolines, Abstracts 18th Fall Workshop on Comput. Geom. (FWCG08), Troy, NY, October 2008.

J. Ferté, V. Pilaud and M. Pocchiola, On the number of simple arrangements of five double pseudolines, arXiv:1009.1575 [cs.CG], 2010; Discrete Comput. Geom. 45 (2011), 279-302.

See A180500 for isomorphism classes of simple arrangements of n pseudolines and m double pseudolines in the projective plane.
See A180502 for isomorphism classes of arrangements of n pseudolines and m double pseudolines in the Moebius strip.
First diagonal gives A063800.

A180503 Triangle read by row. T(n,m) gives the number of isomorphism classes of simple arrangements of n pseudolines and m double pseudolines in the Moebius strip.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 7, 16, 2, 13, 140, 1499, 11502, 3, 122, 5589, 245222, 9186477, 238834187

Offset: 0

Vincent Pilaud, Sep 08 2010

J. Ferté, V. Pilaud and M. Pocchiola, On the number of arrangements of five double pseudolines, Abstracts 18th Fall Workshop on Comput. Geom. (FWCG08), Troy, NY, October 2008.

J. Ferté, V. Pilaud and M. Pocchiola, On the number of simple arrangements of five double pseudolines, arXiv:1009.1575 [cs.CG], 2010; Discrete Comput. Geom. 45 (2011), 279-302.

See A180502 for isomorphism classes of all (not only simple) arrangements of n pseudolines and m double pseudolines in the Moebius strip.
See A180500 for isomorphism classes of simple arrangements of n pseudolines and m double pseudolines in the projective plane.
First diagonal gives A006247.

A180500 Triangle read by row. T(n,m) gives the number of isomorphism classes of simple arrangements of n pseudolines and m double pseudolines in the projective plane.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 4, 13, 1, 5, 48, 626, 6570, 1, 25, 1329, 86715, 4822394, 181403533

Offset: 0

Vincent Pilaud, Sep 08 2010

J. Ferté, V. Pilaud and M. Pocchiola, On the number of arrangements of five double pseudolines, Abstracts 18th Fall Workshop on Comput. Geom. (FWCG08), Troy, NY, October 2008.

J. Ferté, V. Pilaud and M. Pocchiola, On the number of simple arrangements of five double pseudolines, arXiv:1009.1575 [cs.CG], 2010; Discrete Comput. Geom. 45 (2011), 279-302.

See A180501 for isomorphism classes of all (not only simple) arrangements of n pseudolines and m double pseudolines in the projective plane.
See A180503 for isomorphism classes of simple arrangements of n pseudolines and m double pseudolines in the Moebius strip.
First diagonal gives A006248.

A180502 Triangle read by row. T(n,m) gives the number of isomorphism classes of arrangements of n pseudolines and m double pseudolines in the Moebius strip.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 5, 17, 59, 3, 45, 799, 15649, 245351

Offset: 0

Vincent Pilaud, Sep 08 2010

J. Ferté, V. Pilaud and M. Pocchiola, On the number of arrangements of five double pseudolines, Abstracts 18th Fall Workshop on Comput. Geom. (FWCG08), Troy, NY, October 2008.

J. Ferté, V. Pilaud and M. Pocchiola, On the number of simple arrangements of five double pseudolines, arXiv:1009.1575 [cs.CG], 2010; Discrete Comput. Geom. 45 (2011), 279-302.

See A180503 for isomorphism classes of simple arrangements of n pseudolines and m double pseudolines in the Moebius strip.
See A180501 for isomorphism classes of arrangements of n pseudolines and m double pseudolines in the projective plane.
First diagonal gives A063854.

cp's OEIS Frontend

User: Vincent Pilaud

A263791 Number of permutations of [n] avoiding the generalized patterns 1(k+2)-(u_1+1)-...-(u_k+1) for all permutations u of [k].

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A180501 Triangle read by row. T(n,m) gives the number of isomorphism classes of arrangements of n pseudolines and m double pseudolines in the projective plane.

Views

Author

Keywords

References

Links

Crossrefs

A180503 Triangle read by row. T(n,m) gives the number of isomorphism classes of simple arrangements of n pseudolines and m double pseudolines in the Moebius strip.

Views

Author

Keywords

References

Links

Crossrefs

A180500 Triangle read by row. T(n,m) gives the number of isomorphism classes of simple arrangements of n pseudolines and m double pseudolines in the projective plane.

Views

Author

Keywords

References

Links

Crossrefs

A180502 Triangle read by row. T(n,m) gives the number of isomorphism classes of arrangements of n pseudolines and m double pseudolines in the Moebius strip.

Views

Author

Keywords

References

Links

Crossrefs