A000682 -id:A000682 - cp's OEIS frontend

A223094 Number of foldings of n labeled stamps in which leaf n is inwards.

0, 0, 2, 6, 26, 78, 288, 888, 3130, 9850, 34112, 108998, 374636, 1211046, 4148816, 13533796, 46304730, 152153758, 520434552, 1720325302, 5885686496, 19552190624, 66927118548, 223264746520, 764725528072, 2560239468774, 8775478294368, 29470844083770

Offset: 1

N. J. A. Sloane, Mar 29 2013

nonn

Subset of foldings of n labeled stamps (A000136). [Stéphane Legendre, Apr 09 2013]
From Roger Ford, Aug 23 2024: (Start)
a(n) represents the number of impossible stamp foldings with stamp 1 on top and n+1 stamps that are correctly folded for the first n stamps. From stamp n to stamp n+1, the stamp connection crosses a folding so the folding is impossible.
Example a(3) = 2.  Impossible foldings = 1,3,2,4 and 1,4,2,3.
               1 __                      1 __
Stamp numbers  3 __|__   Vertical Lines  4 __|__
               2 |_|  |  lines are folds 2 __| |
               4 _____|                  3 |___|
a(4) = 6, and that means for 5 stamps there are 6 impossible foldings with the first impossible folding occurring from stamp 4 to stamp 5.  Impossible foldings = 1,2,4,3,5; 1,2,5,3,4; 1,3,4,2,5; 1,4,3,5,2; 1,5,2,4,3; 1,5,3,4,2. (End)

Stéphane Legendre, Table of n, a(n) for n = 1..43
Stéphane Legendre, Foldings and Meanders, arXiv preprint arXiv:1302.2025 [math.CO], 2013.
Index entries for sequences obtained by enumerating foldings

Cf. A000136, A000682.

A000682 = Import["https://oeis.org/A000682/b000682.txt", "Table"][[All, 2]];
a[n_] := n A000682[[n]] - A000682[[n + 1]];
Array[a, 43] (* Jean-François Alcover, Sep 02 2019 *)

a(n) = A000136(n) - A000682(n+1). - Andrew Howroyd, Dec 05 2015

For n >= 3: a(n) = n! - Sum_{k=3..n-1} (a(k)*n!/k!) - A000682(n+1). - Roger Ford, Aug 24 2024

More terms from Stéphane Legendre, Apr 09 2013

A046721 Number of semi-meanders of order n with 2 components.

Original entry on oeis.org

1, 2, 6, 16, 48, 140, 428, 1308, 4072, 12796, 40432, 129432, 413900, 1342580, 4335288, 14201804, 46226896, 152594276, 500016036, 1660630740, 5472190206, 18264517264, 60475691308, 202684618564, 673892675030, 2266436498400, 7562707682032, 25510762766704, 85394319699916

Offset: 2

N. J. A. Sloane

nonn

Andrew Howroyd, Table of n, a(n) for n = 2..40
P. Di Francesco, O. Golinelli and E. Guitter, Meander, folding and arch statistics, arXiv:hep-th/9506030, 1995.

Cf. A000682, A046726.

a(15)-a(30) from Andrew Howroyd, Nov 27 2015

A046722 Number of semi-meanders of order n with 3 components.

Original entry on oeis.org

1, 3, 11, 37, 126, 430, 1454, 4976, 16880, 57824, 197010, 675428, 2310268, 7927778, 27205180, 93448486, 321537086, 1105589516, 3812424912, 13121988240, 45330375774, 156172996170, 540314673678, 1863197292582, 6454265995454, 22275589419432, 77246945788890, 266813803179348

Offset: 3

N. J. A. Sloane

nonn

Andrew Howroyd, Table of n, a(n) for n = 3..40
P. Di Francesco, O. Golinelli, and E. Guitter, Meander, folding and arch statistics, arXiv:hep-th/9506030, 1995.
P. Di Francesco, O. Golinelli, and E. Guitter, Meander, folding and arch statistics, Mathematical and Computer Modelling 26 (1997), 97-147.

Cf. A000682, A046726.

a(15)-a(30) from Andrew Howroyd, Nov 27 2015

A046723 Number of semi-meanders of order n with 4 components.

Original entry on oeis.org

1, 4, 17, 66, 254, 956, 3584, 13256, 49052, 179552, 658560, 2394504, 8724464, 31575096, 114451388, 412811544, 1490190544, 5360943684, 19288139802, 69245171564, 248463024330, 890477645192, 3188033497580, 11409453277272, 40771092374710, 145735210316376, 519955750491512

Offset: 4

N. J. A. Sloane

nonn

Andrew Howroyd, Table of n, a(n) for n = 4..40
P. Di Francesco, O. Golinelli and E. Guitter, Meander, folding and arch statistics, arXiv:hep-th/9506030, 1995.

Cf. A000682, A046726.

a(15)-a(40) from Andrew Howroyd, Dec 07 2015

A046724 Number of semi-meanders of order n with 5 components.

Original entry on oeis.org

1, 5, 24, 104, 438, 1796, 7238, 28848, 113518, 444278, 1720384, 6643492, 25421620, 97136712, 368280210, 1395104236, 5250325378, 19746342212, 73863421894, 276113486146, 1027609657470, 3821478801772, 14161346139866, 52428406903688, 193568833452364, 713860635606784

Offset: 5

N. J. A. Sloane

nonn

Andrew Howroyd, Table of n, a(n) for n = 5..40
P. Di Francesco, O. Golinelli and E. Guitter, Meander, folding and arch statistics, arXiv:hep-th/9506030, 1995.

Cf. A000682, A046726.

a(15)-a(40) from Andrew Howroyd, Dec 07 2015

A046725 Number of semi-meanders of order n with 6 components.

Original entry on oeis.org

1, 6, 32, 152, 690, 3028, 12996, 54812, 228284, 939148, 3833076, 15487428, 62244564, 247973928, 984221764, 3876113404, 15223550024, 59379645924, 231124139318, 894157177372, 3453279084296, 13266154255196, 50886266714598, 194294744477756, 740816697816046

Offset: 6

N. J. A. Sloane

nonn

Andrew Howroyd, Table of n, a(n) for n = 6..40
P. Di Francesco, O. Golinelli and E. Guitter, Meander, folding and arch statistics, arXiv:hep-th/9506030, 1995.

Cf. A000682, A046726.

a(15)-a(40) from Andrew Howroyd, Dec 07 2015

A086441 Number of inequivalent ways a semi-infinite curve can cross a straight line n times.

Original entry on oeis.org

1, 1, 2, 4, 11, 27, 79, 213, 644, 1840, 5660

Offset: 1

N. J. A. Sloane, Sep 09 2003

This uses a too broad notion of equivalence. Besides the obvious reflection in a plane perpendicular to the straight line, if the end of the curve is in a free region of the plane, it is extended to infinity and the direction of the curve can then be reversed. A000560 uses a better definition of equivalence.

			The a(3) = 2 solutions with 3 crossings. The line is drawn horizontally. The curve starts at oo and ends at X. The crossings are indicated by stars.
       --        X
      /  \      /
-----*----*----*----
    /      \  /
   /        --
  /
oo
         ---
        /   \
       /  X  \
      /   |   \
-----*----*----*----
    /     |   /
   /      .---
  /
oo

Index entries for sequences obtained by enumerating foldings

Isomorphism classes (using too generous a definition of isomorphism) from A000682. Cf. A000560, A001011.

A223095 Number of foldings of n labeled stamps in which both end leaves are inwards.

Original entry on oeis.org

0, 0, 0, 2, 10, 40, 156, 546, 1986, 6716, 23742, 79472, 277178, 925588, 3205896, 10711486, 36963722, 123712788, 426075994, 1429030624, 4916833424, 16526958144, 56840484232, 191466923584, 658460090994, 2222507917328, 7644360501390, 25850724646008, 88938175307354

Offset: 1

N. J. A. Sloane, Mar 29 2013

nonn

Subset of foldings of n labeled stamps (A000136). - Stéphane Legendre, Apr 09 2013

Stéphane Legendre, Table of n, a(n) for n = 1..42
S. Legendre, Foldings and Meanders, arXiv preprint arXiv:1302.2025 [math.CO], 2013.
S. Legendre, Foldings and Meanders, Aust. J. Comb. 58(2), 275-291, 2014.
Index entries for sequences obtained by enumerating foldings

Cf. A000136, A000682, A077014.

Cf. A223093, A223094.

A217318 = Import["https://oeis.org/A217318/b217318.txt", "Table"][[All, 2]];
a[n_] := (2 - Mod[n, 2]) A217318[[n]];
Array[a, 42] (* Jean-François Alcover, Sep 02 2019 *)

a(n) = A223094(n) - A223093(n). - Andrew Howroyd, Dec 06 2015
a(n) = A000136(n) + A077014(n) - 2 * A000682(n). - Andrew Howroyd, Dec 06 2015
A217318(n) = a(n) if n is odd and A217318(n) = (1/2)*a(n) if n is even. - Stéphane Legendre, Jan 13 2014

Name clarified by Stéphane Legendre, Apr 09 2013

More terms from Stéphane Legendre, Apr 09 2013

A259698 Triangle read by rows: T(n,k) = number of permutations without overlaps having k increasing runs.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 6, 10, 6, 1, 1, 10, 23, 22, 9, 1, 1, 14, 44, 61, 41, 12, 1, 1, 22, 87, 158, 148, 71, 16, 1, 1, 30, 151, 352, 436, 301, 114, 20, 1, 1, 46, 280, 791, 1210, 1092, 589, 175, 25, 1, 1, 62, 464, 1592, 2969, 3317, 2408, 1038, 256, 30, 1

Offset: 2

N. J. A. Sloane, Jul 05 2015

The sums s(n) = Sum_k k*T(n,k) give A259700.

Albert Sade in Sur les Chevauchements des Permutation (published by the author in French in 1949) gave the following example for determining the number of increasing runs in a permutation: 176852943 has 3 runs: 123 (left to right), 34567 (right to left), 789 (right to left).

			Triangle begins:
  1;
  1,  1;
  1,  2,   1;
  1,  4,   4,    1;
  1,  6,  10,    6,    1;
  1, 10,  23,   22,    9,    1;
  1, 14,  44,   61,   41,   12,    1;
  1, 22,  87,  158,  148,   71,   16,    1;
  1, 30, 151,  352,  436,  301,  114,   20,   1;
  1, 46, 280,  791, 1210, 1092,  589,  175,  25,  1;
  1, 62, 464, 1592, 2969, 3377, 2408, 1038, 256, 30, 1;
  ...

Albert Sade, Sur les Chevauchements des Permutations, published by the author, Marseille, 1949. [Annotated scanned copy]

Row sums give A000682.

Cf. A259700.

Overlapfree(v)={for(i=1, #v, for(j=i+1, v[i]-1, if(v[j]>v[i], return(0)))); 1}
Chords(u)={my(n=2*#u, v=vector(n), s=u[#u]); if(s%2==0, s=n+1-s); for(i=1, #u, my(t=n+1-s); s=u[i]; if(s%2==0, s=n+1-s); v[s]=t; v[t]=s); v}
Runs(v)={my(u=vector(#v), s=1); for(i=1, #v, u[v[i]]=i); for(i=2, #u-1, if(sign(u[i]-u[i-1])==sign(u[i]-u[i+1]), s++)); s}
row(n)={my(r=vector(n-1)); if(n>=2, forperm(n, v, if(v[1]<>1, break); if(Overlapfree(Chords(v)), r[Runs(v)]++))); r}
for(n=2, 8, print(row(n))) \\ Andrew Howroyd, Dec 07 2018

Corrected and extended by Roger Ford, Jul 06 2016

A259700 a(n) = Sum_{k=2..n-1} k*A259698(n,k).

Original entry on oeis.org

1, 3, 8, 25, 72, 229, 689, 2224, 6875, 22457, 70767

Offset: 2

N. J. A. Sloane, Jul 05 2015

Sade gives a(5)=25, but that is wrong.

Albert Sade, Sur les Chevauchements des Permutations, published by the author, Marseille, 1949. [Annotated scanned copy]. See Section 14.

Cf. A259698, A000682.

Corrected and extended by Roger Ford, Jul 13 2016

cp's OEIS Frontend

A223094 Number of foldings of n labeled stamps in which leaf n is inwards.

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A046721 Number of semi-meanders of order n with 2 components.

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A046722 Number of semi-meanders of order n with 3 components.

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A046723 Number of semi-meanders of order n with 4 components.

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A046724 Number of semi-meanders of order n with 5 components.

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A046725 Number of semi-meanders of order n with 6 components.

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A086441 Number of inequivalent ways a semi-infinite curve can cross a straight line n times.

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A223095 Number of foldings of n labeled stamps in which both end leaves are inwards.

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Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A259698 Triangle read by rows: T(n,k) = number of permutations without overlaps having k increasing runs.

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Keywords

Comments

Examples

Links

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Programs

PARI

Extensions

A259700 a(n) = Sum_{k=2..n-1} k*A259698(n,k).

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