A334218 -id:A334218 - cp's OEIS frontend

A334778 Triangle read by rows: T(n,k) is the number of permutations of 2 indistinguishable copies of 1..n arranged in a circle with exactly k local maxima.

1, 0, 1, 0, 4, 2, 0, 18, 66, 6, 0, 72, 1168, 1192, 88, 0, 270, 16220, 61830, 33600, 1480, 0, 972, 202416, 2150688, 3821760, 1268292, 40272, 0, 3402, 2395540, 62178928, 272509552, 279561086, 62954948, 1476944, 0, 11664, 27517568, 1629254640, 15313310208, 36381368048, 24342647424, 3963672720, 71865728

Offset: 0

Andrew Howroyd, May 13 2020

T(n,k) is divisible by n for n > 0.

			Triangle begins:
   1;
   0,    1;
   0,    4,       2;
   0,   18,      66,        6;
   0,   72,    1168,     1192,        88;
   0,  270,   16220,    61830,     33600,      1480;
   0,  972,  202416,  2150688,   3821760,   1268292,    40272;
   0, 3402, 2395540, 62178928, 272509552, 279561086, 62954948, 1476944;
  ...
The T(2,1) = 4 permutations of 1122 with 1 local maximum are 1122, 1221, 2112, 2211.
The T(2,2) = 2 permutations of 1122 with 2 local maxima are 1212, 2121.

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Columns k=0..6 are A000007, A027261(n-1), A159716, A159717, A159718, A159719, A159720.
Row sums are A000680.
Main diagonal is A334779.
The version for permutations of 1..n is A263789.
Cf. A334218, A334774, A334780.

CircPeaksBySig(sig, D)={
  my(F(lev,p,q) = my(key=[lev,p,q], z); if(!mapisdefined(FC, key, &z),
    my(m=sig[lev]); z = if(lev==1, if(p==0, binomial(m-1, q), 0), sum(i=0, p, sum(j=0, min(m-i, q), self()(lev-1, p-i, q-j+i) * binomial(m+2*(q-j)+1, 2*q+i-j+1) * binomial(q-j+i, i) * binomial(q+1, j) )));
    mapput(FC, key, z)); z);
  local(FC=Map());
  vector(#D, i, my(k=D[i], lev=#sig); if(lev==1, k==1, my(m=sig[lev]); lev*sum(j=1, min(m,k), m*binomial(m-1,j-1)*F(lev-1,k-j,j-1)/j)));
}
Row(n)={ if(n==0, [1], CircPeaksBySig(vector(n,i,2), [0..n])) }
{ for(n=0, 8, print(Row(n))) }

T(n,k) = n*(2*F(2,n-1,k-1,0) + F(2,n-1,k-2,1)) for n > 1 where F(m,n,p,q) = Sum_{i=0..p} Sum_{j=0..min(m-i, q)} F(m, n-1, p-i, q-j+i) * binomial(m+2*(q-j)+1, 2*q+i-j+1) * binomial(q-j+i, i) * binomial(q+1, j) for n > 1 with F(m,1,0,q) = binomial(m-1, q), F(m,1,p,q) = 0 for p > 0.

A334780(n) = Sum_{k=1..n} k*T(n,k).

A151576 Number of permutations of 1..n arranged in a circle with exactly 3 adjacent element pairs in decreasing order.

Original entry on oeis.org

0, 4, 55, 396, 2114, 9528, 38637, 146080, 526240, 1831644, 6217523, 20716164, 68059710, 221195824, 712856665, 2282058360, 7266358556, 23035517940, 72760054815, 229112753980, 719545590010, 2254604460264, 7050252659525, 22006821057936, 68581455012504, 213411502891468

Offset: 3

R. H. Hardin, May 21 2009

Exactly 2 adjacent element pairs in decreasing order gives A027540(n-1).

Andrew Howroyd, Table of n, a(n) for n = 3..500
Index entries for linear recurrences with constant coefficients, signature (16,-111,438,-1083,1740,-1817,1190,-444,72).

Column k=3 of A334218.
Related sequences: A151577-A151610.
Cf. A000460.

a(n)={n*(3^(n-1) - n*2^(n-1) + n*(n-1)/2)} \\ Andrew Howroyd, May 05 2020

From Andrew Howroyd, May 05 2020: (Start)
a(n) = n*A000460(n-1).
a(n) = n*(3^(n-1) - n*2^(n-1) + n*(n-1)/2).
a(n) = 16*a(n-1) - 111*a(n-2) + 438*a(n-3) - 1083*a(n-4) + 1740*a(n-5) - 1817*a(n-6) + 1190*a(n-7) - 444*a(n-8) + 72*a(n-9).
G.f.: x^4*(4 - 9*x - 40*x^2 + 131*x^3 - 98*x^4)/((1 - x)^4*(1 - 2*x)^3*(1 - 3*x)^2).
(End)

Terms a(18) and beyond from Andrew Howroyd, May 05 2020

A151577 Number of permutations of 1..n arranged in a circle with exactly 4 adjacent element pairs in decreasing order.

Original entry on oeis.org

0, 5, 156, 2114, 19328, 140571, 882340, 5007112, 26441856, 132439905, 637468300, 2976161790, 13569454592, 60725449335, 267757190100, 1166662948900, 5034645823680, 21556696454685, 91704869986620, 388044105102650, 1634678955350400, 6860481786528275, 28700914012807556

Offset: 4

R. H. Hardin, May 21 2009

nonn

Andrew Howroyd, Table of n, a(n) for n = 4..500

Column k=4 of A334218.

Cf. A000498.

a(n) = {n*(4^(n-1) - n*3^(n-1) + binomial(n,2)*2^(n-1) - binomial(n,3))} \\ Andrew Howroyd, May 05 2020

From Andrew Howroyd, May 05 2020: (Start)
a(n) = n*A000498(n-1).
a(n) = n*(4^(n-1) - n*3^(n-1) + binomial(n,2)*2^(n-1) - binomial(n,3)). (End)

Terms a(17) and beyond from Andrew Howroyd, May 05 2020

A151578 Number of permutations of 1..n arranged in a circle with exactly 5 adjacent element pairs in decreasing order.

Original entry on oeis.org

0, 6, 399, 9528, 140571, 1561900, 14413894, 116857368, 862140162, 5925941490, 38576132625, 240659672336, 1451515055333, 8520359419080, 48925419854400, 275923203690000, 1533178869210324, 8414851432723230, 45712442315346915, 246193095207323400, 1316311515774609375

Offset: 5

R. H. Hardin, May 21 2009

nonn

Andrew Howroyd, Table of n, a(n) for n = 5..500

Column k=5 of A334218.

Cf. A000505.

a(n) = {n*(5^(n-1) - n*4^(n-1) + binomial(n,2)*3^(n-1) - binomial(n,3)*2^(n-1) + binomial(n,4))} \\ Andrew Howroyd, May 05 2020

From Andrew Howroyd, May 05 2020: (Start)
a(n) = n*A000505(n-1).
a(n) = n*(5^(n-1) - n*4^(n-1) + binomial(n,2)*3^(n-1) - binomial(n,3)*2^(n-1) + binomial(n,4)). (End)

Terms a(17) and beyond from Andrew Howroyd, May 05 2020

A151579 Number of permutations of 1..n arranged in a circle with exactly 6 adjacent element pairs in decreasing order.

Original entry on oeis.org

0, 7, 960, 38637, 882340, 14413894, 188690976, 2112659718, 21078701112, 192648942945, 1644431982848, 13295963811083, 102911255502876, 768689550213368, 5575887557096640, 39473882067826332, 273820542615005232, 1867156445048432043, 12548621876834960064

Offset: 6

R. H. Hardin, May 21 2009

nonn

Andrew Howroyd, Table of n, a(n) for n = 6..500

Column k=6 of A334218.

Cf. A000514.

a(n) = {n*(6^(n-1) - n*5^(n-1) + binomial(n,2)*4^(n-1) - binomial(n,3)*3^(n-1) + binomial(n,4)*2^(n-1) - binomial(n,5))} \\ Andrew Howroyd, May 05 2020

From Andrew Howroyd, May 05 2020: (Start)
a(n) = n*A000514(n-1).
a(n) = n*(6^(n-1) - n*5^(n-1) + binomial(n,2)*4^(n-1) - binomial(n,3)*3^(n-1) + binomial(n,4)*2^(n-1) - binomial(n,5)). (End)

Terms a(17) and beyond from Andrew Howroyd, May 05 2020

A151580 Number of permutations of 1..n arranged in a circle with exactly 7 adjacent element pairs in decreasing order.

Original entry on oeis.org

0, 8, 2223, 146080, 5007112, 116857368, 2112659718, 31852408056, 419567641380, 4982201574576, 54527214028669, 559001753070984, 5433944411838700, 50558500037520720, 453570522365097900, 3946487691743226960, 33461385366044612088, 277535329177185004536

Offset: 7

R. H. Hardin, May 21 2009

nonn

Andrew Howroyd, Table of n, a(n) for n = 7..500

Column k=7 of A334218.

Cf. A001243.

a(n) = n*A001243(n-1). - Andrew Howroyd, May 05 2020

Terms a(16) and beyond from Andrew Howroyd, May 05 2020

A151581 Number of permutations of 1..n arranged in a circle with exactly 8 adjacent element pairs in decreasing order.

Original entry on oeis.org

0, 9, 5020, 526240, 26441856, 862140162, 21078701112, 419567641380, 7160621079552, 108507581733075, 1496470017336660, 19127489390389332, 229712892700188480, 2619711824193741060, 28608047711740408560, 301155410573256748920, 3072652643940279427584

Offset: 8

R. H. Hardin, May 21 2009

nonn

Andrew Howroyd, Table of n, a(n) for n = 8..500

Column k=8 of A334218.

Cf. A001244.

a(n) = n*A001244(n-1). - Andrew Howroyd, May 05 2020

Terms a(17) and beyond from Andrew Howroyd, May 05 2020

A151582 Number of permutations of 1..n arranged in a circle with exactly 9 adjacent element pairs in decreasing order.

Original entry on oeis.org

0, 10, 11143, 1831644, 132439905, 5925941490, 192648942945, 4982201574576, 108507581733075, 2068026851853900, 35442327497832350, 557245611341867160, 8160353475203019510, 112618455705219419340, 1478355077007310516350, 18597419323476152223600

Offset: 9

R. H. Hardin, May 21 2009

nonn

Andrew Howroyd, Table of n, a(n) for n = 9..500

Column k=9 of A334218.

Terms a(17) and beyond from Andrew Howroyd, May 05 2020

cp's OEIS Frontend

A334778 Triangle read by rows: T(n,k) is the number of permutations of 2 indistinguishable copies of 1..n arranged in a circle with exactly k local maxima.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A151576 Number of permutations of 1..n arranged in a circle with exactly 3 adjacent element pairs in decreasing order.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

Extensions

A151577 Number of permutations of 1..n arranged in a circle with exactly 4 adjacent element pairs in decreasing order.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

Extensions

A151578 Number of permutations of 1..n arranged in a circle with exactly 5 adjacent element pairs in decreasing order.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

Extensions

A151579 Number of permutations of 1..n arranged in a circle with exactly 6 adjacent element pairs in decreasing order.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

Extensions

A151580 Number of permutations of 1..n arranged in a circle with exactly 7 adjacent element pairs in decreasing order.

Views

Author

Keywords

Links

Crossrefs

Formula

Extensions

A151581 Number of permutations of 1..n arranged in a circle with exactly 8 adjacent element pairs in decreasing order.

Views

Author

Keywords

Links

Crossrefs

Formula

Extensions

A151582 Number of permutations of 1..n arranged in a circle with exactly 9 adjacent element pairs in decreasing order.

Views

Author

Keywords

Links

Crossrefs

Extensions