A263789 -id:A263789 - 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).

A159710 Number of permutations of 1..n arranged in a circle with exactly 2 local maxima.

Original entry on oeis.org

0, 0, 0, 0, 8, 80, 528, 2912, 14592, 69120, 316160, 1413632, 6223872, 27103232, 117067776, 502456320, 2145517568, 9122349056, 38644678656, 163186343936, 687144960000, 2886107922432, 12094385684480, 50577004298240, 211105074905088, 879606785638400

Offset: 0

R. H. Hardin, Apr 20 2009

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (14,-76,200,-256,128).

Column k=2 of A263789.

[0,0] cat [2^(-5+n)*(4+2^n-4*n)*n: n in [2..30]]; // G. C. Greubel, Jun 02 2018

LinearRecurrence[{14,-76,200,-256,128},{0,0,0,0,8,80,528},30] (* Harvey P. Dale, Sep 23 2017 *)
Join[{0,0}, Table[2^(-5+n)*(4+2^n-4*n)*n, {n, 2, 30}]] (* G. C. Greubel, Jun 02 2018 *)

concat([0, 0, 0, 0], Vec(-8*(2*x^2-4*x+1)*x^4 / ((4*x-1)^2*(2*x -1)^3) + O(x^100))) \\ Altug Alkan, Oct 26 2015

a(n) = if(n==1, 0, 2^(-5+n)*(4+2^n-4*n)*n) \\ Colin Barker, Oct 26 2015

G.f.: -8*(2*x^2-4*x+1)*x^4 / ((4*x-1)^2*(2*x-1)^3). - Maksym Voznyy (voznyy(AT)mail.ru), Jul 26 2009
a(n) = 2^(-5+n)*(4+2^n-4*n)*n for n>1. - Colin Barker, Oct 26 2015
a(n) = 14*a(n-1) - 76*a(n-2) + 200*a(n-3) - 256*a(n-4) + 128*a(n-5). - Wesley Ivan Hurt, Aug 04 2025

More terms from Alois P. Heinz, Oct 26 2015

A159711 Number of permutations of 1..n arranged in a circle with exactly 3 local maxima.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 96, 1904, 23040, 221184, 1858560, 14353152, 104742912, 734769152, 5010432000, 33464217600, 220066480128, 1430279159808, 9212045819904, 58914039332864, 374665295953920, 2371935399837696, 14960708435312640, 94072038170296320, 589975504803594240

Offset: 0

R. H. Hardin, Apr 20 2009

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (32,-444,3504,-17328,55680,-116288,152320,-113664,36864).

Column k=3 of A263789.

[(1/3)*2^(-6+n)*n*(15+3*2^(1+n)+3^n-3*(8+2^n)*n+6*n^2): n in [0..30]]; // G. C. Greubel, Jun 01 2018

Table[(1/3)*2^(-6+n)*n*(15+3*2^(1+n)+3^n-3*(8+2^n)*n+6*n^2), {n,0,30}] (* G. C. Greubel, Jun 01 2018 *)

a(n) = if(n==1, 0, 1/3*2^(-6+n)*n*(15+3*2^(1+n)+3^n-3*(8+2^n)*n +6*n^2)) \\ Colin Barker, Oct 26 2015

concat(vector(6), Vec(-16*x^6*(144*x^4-444*x^3+296*x^2-73*x+6)/(
(2*x-1)^4*(4*x-1)^3*(6*x-1)^2) + O(x^30))) \\ Colin Barker, Oct 26 2015

G.f.: -16*(144*x^4-444*x^3+296*x^2-73*x+6)*x^6 / ((6*x-1)^2 *(4*x-1)^3 *(2*x-1)^4). - Alois P. Heinz, Oct 26 2015

a(n) = 1/3*2^(-6+n)*n*(15+3*2^(1+n)+3^n-3*(8+2^n)*n+6*n^2) for n>1. - Colin Barker, Oct 26 2015

a(17)-a(24) from Alois P. Heinz, Oct 26 2015

A159712 Number of permutations of 1..n arranged in a circle with exactly 4 local maxima.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 2176, 71424, 1372160, 20252672, 255040512, 2891180032, 30447656960, 303926476800, 2914762424320, 27113686958080, 246327423270912, 2196784154673152, 19305427103907840, 167673167523348480, 1442534103145512960, 12315082531044065280

Offset: 0

R. H. Hardin, Apr 20 2009

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Column k=4 of A263789.

CoefficientList[Series[-128*(34560*x^7 -146880*x^6 +173712*x^5 -97304*x^4 +29808*x^3 -5120*x^2 +462*x-17)*x^8 / ((8*x-1)^2 *(6*x-1)^3 *(4*x-1)^4 *(2*x-1)^5), {x, 0, 50}], x] (* G. C. Greubel, Jun 02 2018 *)

concat(vector(8), Vec(-128*(34560*x^7 -146880*x^6 +173712*x^5 -97304*x^4 +29808*x^3 -5120*x^2 +462*x-17)*x^8 / ((8*x-1)^2 *(6*x-1)^3 *(4*x-1)^4 *(2*x-1)^5) + O(x^100))) \\ Altug Alkan, Oct 26 2015

G.f.: -128*(34560*x^7 -146880*x^6 +173712*x^5 -97304*x^4 +29808*x^3 -5120*x^2 +462*x-17)*x^8 / ((8*x-1)^2 *(6*x-1)^3 *(4*x-1)^4 *(2*x-1)^5). - Alois P. Heinz, Oct 26 2015

a(16)-a(23) from Alois P. Heinz, Oct 26 2015

A159713 Number of permutations of 1..n arranged in a circle with exactly 5 local maxima.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 79360, 3891712, 108736512, 2283154432, 40155709440, 625974681600, 8946380963840, 119830778347520, 1527173964103680, 18720292422287360, 222492157815029760, 2579416038567051264, 29306002590306140160, 327494389862875791360

Offset: 0

R. H. Hardin, Apr 20 2009

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Column k=5 of A263789.

concat(vector(10), Vec(-512*(232243200*x^11 -1275402240*x^10 +2252081664*x^9 -2074564992*x^8 +1174193888*x^7 -439150208*x^6 +112057808*x^5 -19636984*x^4 +2325314*x^3 -177676*x^2 +7899*x -155)*x^10 / ((10*x-1)^2 *(8*x-1)^3 *(6*x-1)^4 *(4*x-1)^5 *(2*x-1)^6) + O(x^100))) \\ Altug Alkan, Oct 26 2015

G.f.: -512*(232243200*x^11 -1275402240*x^10 +2252081664*x^9 -2074564992*x^8 +1174193888*x^7 -439150208*x^6 +112057808*x^5 -19636984*x^4 +2325314*x^3 -177676*x^2 +7899*x -155)*x^10 / ((10*x-1)^2 *(8*x-1)^3 *(6*x-1)^4 *(4*x-1)^5 *(2*x-1)^6). - Alois P. Heinz, Oct 26 2015

a(16)-a(23) from Alois P. Heinz, Oct 26 2015

A159714 Number of permutations of 1..n arranged in a circle with exactly 6 local maxima.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4245504, 290787328, 11134212096, 315250053120, 7373732315136, 151048265662464, 2807359026757632, 48456016702472192, 789426139189739520, 12282937010848530432, 184138764390344687616, 2677761622120892203008

Offset: 0

R. H. Hardin, Apr 20 2009

Alois P. Heinz, Table of n, a(n) for n = 0..930

Column k=6 of A263789.

concat(vector(12), Vec(-2048*(24078974976000*x^16 -163737029836800*x^15 +392749501317120*x^14 -517950785912832*x^13 +442017305468928*x^12 -264568164065280*x^11 +116194714660608*x^10 -38443271058176*x^9 +9722233013888*x^8 -1890674565824*x^7 +282315254112*x^6 -32071886064*x^5 +2720304072*x^4 -166678732*x^3 +6962515*x^2 -177256*x +2073)*x^12 / ((12*x-1)^2 *(10*x-1)^3 *(8*x-1)^4 *(6*x-1)^5 *(4*x-1)^6 *(2*x-1)^7) + O(x^100))) \\ Altug Alkan, Oct 26 2015

G.f.: -2048*(24078974976000*x^16 -163737029836800*x^15 +392749501317120*x^14 -517950785912832*x^13 +442017305468928*x^12 -264568164065280*x^11 +116194714660608*x^10 -38443271058176*x^9 +9722233013888*x^8 -1890674565824*x^7 +282315254112*x^6 -32071886064*x^5 +2720304072*x^4 -166678732*x^3 +6962515*x^2 -177256*x +2073)*x^12 / ((12*x-1)^2 *(10*x-1)^3 *(8*x-1)^4 *(6*x-1)^5 *(4*x-1)^6 *(2*x-1)^7). - Alois P. Heinz, Oct 26 2015

a(17)-a(23) from Alois P. Heinz, Oct 26 2015

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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A159710 Number of permutations of 1..n arranged in a circle with exactly 2 local maxima.

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A159711 Number of permutations of 1..n arranged in a circle with exactly 3 local maxima.

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A159712 Number of permutations of 1..n arranged in a circle with exactly 4 local maxima.

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A159713 Number of permutations of 1..n arranged in a circle with exactly 5 local maxima.

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PARI

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A159714 Number of permutations of 1..n arranged in a circle with exactly 6 local maxima.

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Author

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Programs

PARI

Formula

Extensions