A001181 -id:A001181 - cp's OEIS frontend

A363910 Triangular array read by rows: T(n,k) = the number of closed meanders with n top arches and k closed meanders in the reduction of the closed meander by the reverse of the exterior arch splitting algorithm.

1, 0, 2, 0, 2, 6, 0, 6, 14, 22, 0, 28, 56, 86, 92, 0, 162, 298, 428, 518, 422, 0, 1076, 1868, 2562, 3096, 3144, 2074, 0, 7852, 13076, 17292, 20624, 21990, 19366, 10754

Offset: 1

Roger Ford, Jun 27 2023

The terms of this sequence can also be derived from sequences of consecutively numbered stamps folded with stamp 1 on top.

			n\k  1     2      3     4     5     6     7     8
1:   1
2:   0     2
3:   0     2      6
4:   0     6     14    22
5:   0    28     56    86    92
6:   0   162    298   428   518   422
7:   0  1076   1868  2562  3096  3144  2074
8:   0  7852  13076 17292 20624 21990 19366 10754
Closed meander:         Closed meander split with bottom rotated right
4 top arches            to form top of semi-meander with 8 arches
    ______                   ______
   / ____ \                 / ____ \
  / / __ \ \               / / __ \ \              __
 / / /  \ \ \             / / /  \ \ \            /  \
/ / / /\ \ \ \           / / / /\ \ \ \  /\  /\  / /\ \
\ \/ /  \/  \/           binary representation of semi-meander
 \__/                    1 1 1 1 0 0 0 0 1 0 1 0 1 1 0 0
                    Semi-meander top arches with no covering center arch  =  cm
                              START:          center |
Reduction of semi-meander:    1  1  1  1  0  0  0  0  1  0  1  0  1  1  0  0 cm(1)
Combine end of first arch     1  1  1  1  0  0  0  0e 1  0  1  0  1s 1  0  0
Oe with beginning of last        1  1  1  0  0  0  1  1  0  1  0  0  1  0
arch 1s.  0e...1s becomes        1  1  1  0  0  0e 1  1  0  1  0  0  1s 0
1...0 in the next line. The         1  1  0  0  1  1  1  0  1  0  0  0
starting 1 and ending 0             1  1  0  0e 1s 1  1  0  1  0  0  0
are removed in the next line           1  0  1  0  1  1  0  1  0  0
reducing number of arches.             1  0e 1  0  1s 1  0  1  0
by one.                                   1  1  0  0  1  0  1  0             cm(2)
                                          1  1  0  0e 1  0  1s 0
                                             1  0  1  1  0  0
                                             1  0e 1s 1  0  0
                                                1  0  1  0                   cm(3)
  Example: T(4,3) 4 starting top arches with 3 closed meanders in history.

Cf. A005315 (row sums), A001181, A005316, A000682.

T(n,n) = A001181(n).

T(n,2) = A005316(2*n-4)*2 for n > 1.

A375913 Number of strong (=generic) guillotine rectangulations with n rectangles.

Original entry on oeis.org

1, 2, 6, 24, 114, 606, 3494, 21434, 138100, 926008, 6418576, 45755516, 334117246, 2491317430, 18919957430, 146034939362, 1143606856808, 9072734766636, 72827462660824, 590852491725920, 4840436813758832, 40009072880216344, 333419662183186932, 2799687668599080296

Offset: 1

Andrei Asinowski, Sep 02 2024

nonn

Equivalently: The number of strong rectangulations with n rectangles that avoid two windmill patterns.

Andrei Asinowski, Jean Cardinal, Stefan Felsner, and Éric Fusy, Combinatorics of rectangulations: Old and new bijections, arXiv:2402.01483 [math.CO], 2024, page 37.
Arturo Merino and Torsten Mütze, Combinatorial generation via permutation languages. III. Rectangulations, Discrete Comput. Geom., 70(1):51-122, 2023. Page 99, Table 3, entry "12".

Cf. A342141 (number of strong (=generic) rectangulations).
Cf. A001181 (Baxter numbers: number of weak (=diagonal) rectangulations).
Cf. A006318 (Schröder numbers: number of weak (=diagonal) guillotine rectangulations).

A 5-variate recurrence is given in the paper Asinowski, Cardinal, Felsner, and Fusy.

A375923 Number of permutations of size n which are both two-clumped and co-two-clumped.

Original entry on oeis.org

1, 1, 2, 6, 24, 112, 582, 3272, 19550, 122628, 800392, 5400342, 37475474, 266412680, 1934033968, 14300538652, 107471798112, 819442325086, 6329551390064, 49465665347580, 390692732060804, 3115700976866356, 25067250869113332, 203317147838575616, 1661425311693158000

Offset: 0

Andrei Asinowski, Sep 02 2024

nonn

Two-clumped permutations are (3-51-2-4, 3-51-4-2, 2-4-51-3, 4-2-51-3)-avoiding permutations. Co-two-clumped permutations are (3-15-2-4, 3-15-4-2, 2-4-15-3, 4-2-15-3)-avoiding permutations. Thus, this sequence enumerates permutations that avoid all these eight patterns.

a(n) is also the number of strong (=generic) rectangulations of size n whose strong poset is totally ordered.

Andrei Asinowski, Jean Cardinal, Stefan Felsner, and Éric Fusy, Combinatorics of rectangulations: Old and new bijections, arXiv:2402.01483 [math.CO], 2024, page 29.

Cf. A342141 (number of two-clumped permutations).
Cf. A001181 (Baxter numbers: number of (twisted-)Baxter permutations).
Cf. A348351 (number of permutations which are both twisted-Baxter and co-twisted-Baxter).

A383372 Number of centrally symmetric Baxter permutations of length n.

Original entry on oeis.org

1, 1, 2, 2, 6, 8, 26, 38, 130, 202, 712, 1152, 4144, 6904, 25202, 42926, 158442, 274586, 1022348, 1796636, 6736180, 11974360, 45154320, 81040720, 307069360, 555620080, 2113890560, 3851817920, 14705955008, 26960013552, 103245460226

Offset: 0

Ludovic Schwob, Apr 24 2025

nonn

For all n > 0, a(n) is the number of triples of non-intersecting lattice paths of length n-1.

a(n) is the number of symmetric twin pairs of full binary trees with n internal nodes.

			The Baxter permutations corresponding to a(4) = 6 are 1234, 1324, 2143, 3412, 4231, and 4321.

Stefan Felsner, Eric Fusy, Marc Noy, and David Orden, Bijections for Baxter families and related objects, J. Combin. Theory Ser. A, 118(3):993-1020, 2011.
Kevin Dilks, Involutions on Baxter Objects, arXiv:1402.2961 [math.CO], 2014.

Cf. A001181.

For all n>0, a(n) = Sum_{k=0...n-1} Theta_{k,n-k-1}, where Theta_{k,l} is equal to:
  - C(a+b+1,a+1)*C(a+b+1,a)*C(a+b,a)/(a+b+1) if k and l are even with k = 2*a and l = 2*b;
  - C(a+b+1,a+1)^2*C(a+b+1,a)/(a+b+1) if k is odd and l is even with k = 2*a+1 and l = 2*b;
  - Theta(l,k) if k is even and l is odd;
  - 0 if k and l are odd.

cp's OEIS Frontend

A363910 Triangular array read by rows: T(n,k) = the number of closed meanders with n top arches and k closed meanders in the reduction of the closed meander by the reverse of the exterior arch splitting algorithm.

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A375913 Number of strong (=generic) guillotine rectangulations with n rectangles.

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A375923 Number of permutations of size n which are both two-clumped and co-two-clumped.

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A383372 Number of centrally symmetric Baxter permutations of length n.

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