A159716 -id:A159716 - 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).

A334772 Array read by antidiagonals: T(n,k) is the number of permutations of k indistinguishable copies of 1..n arranged in a circle with exactly 2 local maxima.

Original entry on oeis.org

2, 12, 66, 36, 576, 1168, 80, 2610, 17376, 16220, 150, 8520, 129800, 448800, 202416, 252, 22680, 659560, 5748750, 10861056, 2395540, 392, 52416, 2596608, 46412200, 241987500, 253940736, 27517568, 576, 109116, 8505728, 273322980, 3121135440, 9885006250, 5807161344, 310123764

Offset: 2

Andrew Howroyd, May 10 2020

T(n,k) is divisible by n and 2*T(n,k) is divisible by n*k.

			Array begins:
==========================================================
n\k |        2          3            4              5
----|----------------------------------------------------
  2 |        2         12           36             80 ...
  3 |       66        576         2610           8520 ...
  4 |     1168      17376       129800         659560 ...
  5 |    16220     448800      5748750       46412200 ...
  6 |   202416   10861056    241987500     3121135440 ...
  7 |  2395540  253940736   9885006250   203933233280 ...
  8 | 27517568 5807161344 395426250000 13051880894720 ...
...
The T(2,3) = 12 permutations of 111222 with 2 local maxima are 112122, 112212 and their rotations.
The T(3,2) = 66 permutations of 112233 with 2 local maxima are 112323, 113223, 113232, 121233, 121332, 122133, 122313, 123213, 123123, 123132, 131322 and their rotations.

Andrew Howroyd, Table of n, a(n) for n = 2..1276

Columns k=2..6 are A159716, A159722, A159728, A159734, A159737.

Cf. A334773, A334774, A334778.

T(n,k)={n*k*( (k^2 + 4*k + 1)^2*binomial(k+3,3)^(n-2) + 12*(k + 2)*(k+1)^(n-2) - 6*k*(k+5)*n*(k+1)^(n-2))/(2*(k + 5)^2)}

T(n,k) = n*k*( P(k,4)^(n-2) * P(k-2,2)^2 + 4*(Sum_{j=0..n-3} P(k-1,3) * P(k-2,2) * P(k,2)^j * P(k, 4)^(n-j-3)) + 4*(Sum_{j=0..n-4} (j + 1) * P(k-1,3)^2 * P(k,2)^j * P(k,4)^(n-j-4)) )/2 where P(n,k) = binomial(n+k-1, k-1).

T(n,k) = n*k*( (k^2 + 4*k + 1)^2*binomial(k+3, 3)^(n-2) + 12*(k + 2)*(k+1)^(n-2) - 6*k*(k+5)*n*(k+1)^(n-2))/(2*(k + 5)^2).

A159722 Number of permutations of 3 indistinguishable copies of 1..n arranged in a circle with exactly 2 local maxima.

Original entry on oeis.org

12, 576, 17376, 448800, 10861056, 253940736, 5807161344, 130675728384, 2903978803200, 63887897001984, 1393919508086784, 30201597684350976, 650495989232173056, 13939199950454784000, 297369599774111563776, 6319103998978368208896, 133816319995412169621504

Offset: 2

R. H. Hardin, Apr 20 2009

Andrew Howroyd, Table of n, a(n) for n = 2..200
Index entries for linear recurrences with constant coefficients, signature (52,-928,6784,-21760,25600).

Column k=3 of A334772.

Cf. A159716.

a(n) = {3*n*(121*20^(n-2) + 15*4^(n-2) - 36*n*4^(n-2))/32} \\ Andrew Howroyd, May 10 2020

Vec(12*x*(1 + 2*x)*(1 - 6*x - 108*x^2 + 80*x^3) / ((1 - 4*x)^3*(1 - 20*x)^2) + O(x^20)) \\ Colin Barker, Jul 16 2020

a(n) = 3*n*(121*20^(n-2) + 15*4^(n-2) - 36*n*4^(n-2))/32. - Andrew Howroyd, May 10 2020
From Colin Barker, Jul 16 2020: (Start)
G.f.: 12*x*(1 + 2*x)*(1 - 6*x - 108*x^2 + 80*x^3) / ((1 - 4*x)^3*(1 - 20*x)^2).
a(n) = 52*a(n-1) - 928*a(n-2) + 6784*a(n-3) - 21760*a(n-4) + 25600*a(n-5) for n>6.
(End)

Terms a(9) and beyond from Andrew Howroyd, May 09 2020

A159728 Number of permutations of 4 indistinguishable copies of 1..n arranged in a circle with exactly 2 local maxima.

Original entry on oeis.org

36, 2610, 129800, 5748750, 241987500, 9885006250, 395426250000, 15570077343750, 605504070312500, 23311913238281250, 890091272109375000, 33749294301074218750, 1272088786561523437500, 47703329503967285156250, 1780924301526757812500000, 66228122463283630371093750

Offset: 2

R. H. Hardin, Apr 20 2009

Andrew Howroyd, Table of n, a(n) for n = 2..200
Index entries for linear recurrences with constant coefficients, signature (85,-2350,23750,-100625,153125).

Column k=4 of A334772.

Cf. A159716.

a(n) = {2*n*(121*35^(n-2) + 8*5^(n-2) - 24*n*5^(n-2))/9} \\ Andrew Howroyd, May 10 2020

Vec(2*x^2*(3 + 5*x)*(6 - 85*x - 1100*x^2 + 875*x^3) / ((1 - 5*x)^3*(1 - 35*x)^2) + O(x^40)) \\ Colin Barker, Jul 16 2020

a(n) = 2*n*(121*35^(n-2) + 8*5^(n-2) - 24*n*5^(n-2))/9. - Andrew Howroyd, May 10 2020

Terms a(8) and beyond from Andrew Howroyd, May 09 2020

A159734 Number of permutations of 5 indistinguishable copies of 1..n arranged in a circle with exactly 2 local maxima.

Original entry on oeis.org

80, 8520, 659560, 46412200, 3121135440, 203933233280, 13051880894720, 822269693093760, 51163456598214400, 3151668992962800640, 192538324414433556480, 11680658351228331345920, 704433549821153777192960, 42266012989435750480281600, 2524689842570106278817955840

Offset: 2

R. H. Hardin, Apr 20 2009

Andrew Howroyd, Table of n, a(n) for n = 2..200
Index entries for linear recurrences with constant coefficients, signature (130,-5260,68760,-362880,677376).

Column k=5 of A334772.

Cf. A159716.

a(n) = {n*(23^2*56^(n-2) + 21*6^(n-2) - 75*n*6^(n-2))/10} \\ Andrew Howroyd, May 10 2020

Vec(40*x^2*(2 + 3*x)*(1 - 25*x - 303*x^2 + 252*x^3) / ((1 - 6*x)^3*(1 - 56*x)^2) + O(x^18)) \\ Colin Barker, Jul 16 2020

a(n) = n*(23^2*56^(n-2) + 21*6^(n-2) - 75*n*6^(n-2))/10. - Andrew Howroyd, May 10 2020
From Colin Barker, Jul 16 2020: (Start)
G.f.: 40*x^2*(2 + 3*x)*(1 - 25*x - 303*x^2 + 252*x^3) / ((1 - 6*x)^3*(1 - 56*x)^2).
a(n) = 130*a(n-1) - 5260*a(n-2) + 68760*a(n-3) - 362880*a(n-4) + 677376*a(n-5) for n>6.
(End)

Terms a(7) and beyond from Andrew Howroyd, May 09 2020

A159717 Number of permutations of 2 indistinguishable copies of 1..n arranged in a circle with exactly 3 local maxima.

Original entry on oeis.org

0, 0, 6, 1192, 61830, 2150688, 62178928, 1629254640, 40346856234, 965510596600, 22606163844396, 521603874280248, 11911230805813846, 269907065756299440, 6079103449024019880, 136243494317831152480, 3040751938796332410018, 67621304208554979697224, 1499043510801269678080708

Offset: 1

R. H. Hardin, Apr 20 2009

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..200

Column 3 of A334778.

Cf. A159716.

\\ CircPeaksBySig defined in A334778.
a(n) = {CircPeaksBySig(vector(n, i, 2), [3])[1]} \\ Andrew Howroyd, May 13 2020

Terms a(11) and beyond from Andrew Howroyd, May 13 2020

A159718 Number of permutations of 2 indistinguishable copies of 1..n arranged in a circle with exactly 4 local maxima.

Original entry on oeis.org

0, 0, 0, 88, 33600, 3821760, 272509552, 15313310208, 750469872312, 33813251867920, 1443455210369040, 59454199364673024, 2389923754993613176, 94450458835284703536, 3687585353084799432720, 142691482885508987276800, 5484263653598164634676600, 209677462059979688650122960

Offset: 1

R. H. Hardin, Apr 20 2009

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..200

Column 4 of A334778.

Cf. A159716.

\\ CircPeaksBySig defined in A334778.
a(n) = {CircPeaksBySig(vector(n, i, 2), [4])[1]} \\ Andrew Howroyd, May 13 2020

Terms a(10) and beyond from Andrew Howroyd, May 13 2020

A159719 Number of permutations of 2 indistinguishable copies of 1..n arranged in a circle with exactly 5 local maxima.

Original entry on oeis.org

0, 0, 0, 0, 1480, 1268292, 279561086, 36381368048, 3573883594170, 296395007981680, 22044296362400136, 1523944523765510064, 100158396249221188476, 6351609408030664973692, 392562103869990035520330, 23810390333486683269302048, 1424190819067621511845096358

Offset: 1

R. H. Hardin, Apr 20 2009

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..200

Column 5 of A334778.

Cf. A159716.

\\ CircPeaksBySig defined in A334778.
a(n) = {CircPeaksBySig(vector(n, i, 2), [5])[1]} \\ Andrew Howroyd, May 13 2020

Terms a(10) and beyond from Andrew Howroyd, May 13 2020

A159720 Number of permutations of 2 indistinguishable copies of 1..n arranged in a circle with exactly 6 local maxima.

Original entry on oeis.org

0, 0, 0, 0, 0, 40272, 62954948, 24342647424, 5320007368884, 848044852469680, 111078667024032048, 12769013592631944576, 1340902091662029846456, 132008300342568131914656, 12398363733385845967412220, 1124539850663707285433353472, 99357839137277548804214431980

Offset: 1

R. H. Hardin, Apr 20 2009

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..200

Column 6 of A334778.

Cf. A159716.

\\ CircPeaksBySig defined in A334778.
a(n) = {CircPeaksBySig(vector(n, i, 2), [6])[1]} \\ Andrew Howroyd, May 13 2020

Terms a(10) and beyond from Andrew Howroyd, May 13 2020

A159737 Number of permutations of 6 indistinguishable copies of 1..n arranged in a circle with exactly 2 local maxima.

Original entry on oeis.org

150, 22680, 2596608, 273322980, 27558217008, 2700777267972, 259275295383552, 24501521550788100, 2286808732032093360, 211301127303186249252, 19362866942233277773632, 1762020891775616889450852, 159395120671659354639719856, 14345560860451487040265198020

Offset: 2

R. H. Hardin, Apr 20 2009

Andrew Howroyd, Table of n, a(n) for n = 2..200
Index entries for linear recurrences with constant coefficients, signature (189,-10731,173215,-1094856,2420208).

Column k=6 of A334772.

Cf. A159716.

a(n) = {3*n*(61^2*84^(n-2) + 96*7^(n-2) - 396*n*7^(n-2))/121} \\ Andrew Howroyd, May 10 2020

Vec(6*x^2*(5 + 7*x)*(5 - 196*x - 2401*x^2 + 2058*x^3) / ((1 - 7*x)^3*(1 - 84*x)^2) + O(x^40)) \\ Colin Barker, Jul 18 2020

a(n) = 3*n*(61^2*84^(n-2) + 96*7^(n-2) - 396*n*7^(n-2))/121. - Andrew Howroyd, May 10 2020
From Colin Barker, Jul 18 2020: (Start)
G.f.: 6*x^2*(5 + 7*x)*(5 - 196*x - 2401*x^2 + 2058*x^3) / ((1 - 7*x)^3*(1 - 84*x)^2).
a(n) = 189*a(n-1) - 10731*a(n-2) + 173215*a(n-3) - 1094856*a(n-4) + 2420208*a(n-5) for n>6.
(End)

Terms a(7) and beyond from Andrew Howroyd, May 09 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.

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A334772 Array read by antidiagonals: T(n,k) is the number of permutations of k indistinguishable copies of 1..n arranged in a circle with exactly 2 local maxima.

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A159722 Number of permutations of 3 indistinguishable copies of 1..n arranged in a circle with exactly 2 local maxima.

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A159728 Number of permutations of 4 indistinguishable copies of 1..n arranged in a circle with exactly 2 local maxima.

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A159734 Number of permutations of 5 indistinguishable copies of 1..n arranged in a circle with exactly 2 local maxima.

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A159717 Number of permutations of 2 indistinguishable copies of 1..n arranged in a circle with exactly 3 local maxima.

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A159718 Number of permutations of 2 indistinguishable copies of 1..n arranged in a circle with exactly 4 local maxima.

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A159719 Number of permutations of 2 indistinguishable copies of 1..n arranged in a circle with exactly 5 local maxima.

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