A334773 -id:A334773 - cp's OEIS frontend

A334774 Triangle read by rows: T(n,k) is the number of permutations of 2 indistinguishable copies of 1..n with exactly k local maxima.

1, 3, 3, 9, 57, 24, 27, 705, 1449, 339, 81, 7617, 48615, 49695, 7392, 243, 78357, 1290234, 3650706, 2234643, 230217, 729, 791589, 30630618, 197457468, 314306943, 128203119, 9689934, 2187, 7944321, 686779323, 9080961729, 30829608729, 31435152267, 9159564513, 529634931

Offset: 1

Andrew Howroyd, May 11 2020

Also the number of permutations of 2 indistinguishable copies of 1..n with exactly k-1 peaks. A peak is an interior maximum.

			Triangle begins:
    1;
    3,      3;
    9,     57,       24;
   27,    705,     1449,       339;
   81,   7617,    48615,     49695,      7392;
  243,  78357,  1290234,   3650706,   2234643,    230217;
  729, 791589, 30630618, 197457468, 314306943, 128203119, 9689934;
  ...
The T(2,1) = 3 permutations of 1122 with 1 local maxima are 1122, 1221, 2211.
The T(2,2) = 3 permutations of 1122 with 2 local maxima are 1212, 2112, 2121.
The T(2,1) = 3 permutations of 1122 with 0 peaks are 2211, 2112, 1122.
The T(2,2) = 3 permutations of 1122 with 1 peak are 2121, 1221, 1212.

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Columns k=1..6 are A000244(n-1), 3*A152494, 3*A152495, 3*A152496, 3*A152497, 3*A152498.
Row sums are A000680.
Main diagonal is A334775.
The version for permutations of 1..n is A008303(n,k-1).
Cf. A154283, A334773, A334776, A334777, A334778.

PeaksBySig(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, F(#sig, D[i], 0));
}
Row(n)={ PeaksBySig(vector(n,i,2), [0..n-1]) }
{ for(n=1, 8, print(Row(n))) }

T(n,k) = F(2,n,k-1,0) 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.
A334776(n) = Sum_{k=1..n} (k-1)*T(n,k).
A334777(n) = Sum_{k=1..n} k*T(n,k).

A152494 1/3 of the number of permutations of 2 indistinguishable copies of 1..n with exactly 2 local maxima.

Original entry on oeis.org

0, 1, 19, 235, 2539, 26119, 263863, 2648107, 26513875, 265250287, 2652876847, 26530008499, 265304159371, 2653054879735, 26530591844071, 265306057146811, 2653061016284227, 26530611583384063, 265306120353746335, 2653061217872021443, 26530612224048411643

Offset: 1

R. H. Hardin, Dec 06 2008

Andrew Howroyd, Table of n, a(n) for n = 1..200
Index entries for linear recurrences with constant coefficients, signature (16,-69,90).

Cf. A334773.

a(n) = {(13*10^(n-1) - 13*3^(n-1) - 14*(n-1)*3^(n-1))/49} \\ Andrew Howroyd, May 10 2020

concat(0, Vec(x*(1 + 3*x) / ((1 - 3*x)^2*(1 - 10*x)) + O(x^20))) \\ Colin Barker, May 19 2020

a(n) = (13*10^(n-1) - 13*3^(n-1) - 14*(n-1)*3^(n-1))/49. - Andrew Howroyd, May 10 2020
From Colin Barker, May 19 2020: (Start)
G.f.: x*(1 + 3*x) / ((1 - 3*x)^2*(1 - 10*x)).
a(n) = 16*a(n-1) - 69*a(n-2) + 90*a(n-3) for n>3.
(End)

Terms a(12) and beyond from Andrew Howroyd, May 10 2020

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).

A152499 1/12 of the number of permutations of 3 indistinguishable copies of 1..n with exactly 2 local maxima.

Original entry on oeis.org

0, 1, 30, 664, 13632, 274432, 5497344, 109987840, 2199945216, 43999756288, 879998926848, 17599995314176, 351999979683840, 7039999912443904, 140799999624609792, 2815999998397775872, 56319999993188450304, 1126399999971143188480, 22527999999878130302976

Offset: 1

R. H. Hardin, Dec 06 2008

Andrew Howroyd, Table of n, a(n) for n = 1..200
Index entries for linear recurrences with constant coefficients, signature (28,-176,320).

Cf. A152494, A334773.

a(n) = {(11*20^(n-1) - 11*4^(n-1) - 12*(n-1)*4^(n-1))/128} \\ Andrew Howroyd, May 10 2020

concat(0, Vec(x^2*(1 + 2*x) / ((1 - 4*x)^2*(1 - 20*x)) + O(x^40))) \\ Colin Barker, Jul 15 2020

a(n) = (11*20^(n-1) - 11*4^(n-1) - 12*(n-1)*4^(n-1))/128. - Andrew Howroyd, May 10 2020
From Colin Barker, Jul 15 2020: (Start)
G.f.: x^2*(1 + 2*x) / ((1 - 4*x)^2*(1 - 20*x)).
a(n) = 28*a(n-1) - 176*a(n-2) + 320*a(n-3) for n>2.
(End)

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

A152504 1/10 of the number of permutations of 4 indistinguishable copies of 1..n with exactly 2 local maxima.

Original entry on oeis.org

0, 3, 140, 5175, 183000, 6416875, 224662500, 7863609375, 275228750000, 9633019921875, 337155773437500, 11800452490234375, 413015839453125000, 14455554393310546875, 505944403833007812500, 17708054134515380859375, 619781894709960937500000, 21692366314858856201171875

Offset: 1

R. H. Hardin, Dec 06 2008

Andrew Howroyd, Table of n, a(n) for n = 1..200
Index entries for linear recurrences with constant coefficients, signature (45,-375,875).

Cf. A152494, A334773.

a(n) = {(11*35^(n-1) - 11*5^(n-1) - 12*(n-1)*5^(n-1))/90} \\ Andrew Howroyd, May 10 2020

concat(0, Vec(x^2*(3 + 5*x) / ((1 - 5*x)^2*(1 - 35*x)) + O(x^20))) \\ Colin Barker, Jul 16 2020

a(n) = (11*35^(n-1) - 11*5^(n-1) - 12*(n-1)*5^(n-1))/90. - Andrew Howroyd, May 10 2020
From Colin Barker, Jul 16 2020: (Start)
G.f.: x^2*(3 + 5*x) / ((1 - 5*x)^2*(1 - 35*x)).
a(n) = 45*a(n-1) - 375*a(n-2) + 875*a(n-3) for n>3.
(End)

Terms a(9) and beyond from Andrew Howroyd, May 10 2020

A152509 1/30 of the number of permutations of 5 indistinguishable copies of 1..n with exactly 2 local maxima.

Original entry on oeis.org

0, 2, 139, 8036, 452068, 25331360, 1418668912, 79446252224, 4448995583296, 249143789616128, 13952052465406720, 781314939695363072, 43753636633642845184, 2450203651553656365056, 137211404487455350386688, 7683838651300399095726080, 430294964472840921667551232

Offset: 1

R. H. Hardin, Dec 06 2008

Andrew Howroyd, Table of n, a(n) for n = 1..200
Index entries for linear recurrences with constant coefficients, signature (68,-708,2016).

Cf. A152494, A334773.

LinearRecurrence[{68,-708,2016},{0,2,139},20] (* Harvey P. Dale, Feb 03 2022 *)

a(n) = {(23*56^(n-1) - 23*6^(n-1) - 25*(n-1)*6^(n-1))/500} \\ Andrew Howroyd, May 10 2020

concat(0, Vec(x^2*(2 + 3*x) / ((1 - 6*x)^2*(1 - 56*x)) + O(x^20))) \\ Colin Barker, Jul 16 2020

a(n) = (23*56^(n-1) - 23*6^(n-1) - 25*(n-1)*6^(n-1))/500. - Andrew Howroyd, May 10 2020
From Colin Barker, Jul 16 2020: (Start)
G.f.: x^2*(2 + 3*x) / ((1 - 6*x)^2*(1 - 56*x)).
a(n) = 68*a(n-1) - 708*a(n-2) + 2016*a(n-3) for n>3.
(End)

Terms a(8) and beyond from Andrew Howroyd, May 10 2020

A152513 1/21 of the number of permutations of 6 indistinguishable copies of 1..n with exactly 2 local maxima.

Original entry on oeis.org

0, 5, 497, 42581, 3584693, 301183841, 25300030889, 2125207418285, 178517461842461, 14995467100301177, 1259619238806161681, 105808016078078472389, 8887873350698981879429, 746581361459780256986513, 62712834362629583374730873, 5267878086460945365330876893

Offset: 1

R. H. Hardin, Dec 06 2008

Andrew Howroyd, Table of n, a(n) for n = 1..200
Index entries for linear recurrences with constant coefficients, signature (98,-1225,4116).

Cf. A152494, A334773.

a(n) = {(61*84^(n-1) - 61*7^(n-1) - 66*(n-1)*7^(n-1))/847} \\ Andrew Howroyd, May 10 2020

Vec(x^2*(5 + 7*x) / ((1 - 7*x)^2*(1 - 84*x)) + O(x^18)) \\ Colin Barker, Jul 16 2020

a(n) = (61*84^(n-1) - 61*7^(n-1) - 66*(n-1)*7^(n-1))/847. - Andrew Howroyd, May 10 2020
From Colin Barker, Jul 16 2020: (Start)
G.f.: x^2*(5 + 7*x) / ((1 - 7*x)^2*(1 - 84*x)).
a(n) = 98*a(n-1) - 1225*a(n-2) + 4116*a(n-3) for n>3.
(End)

Terms a(7) and beyond from Andrew Howroyd, May 10 2020

A152517 1/56 of the number of permutations of 7 indistinguishable copies of 1..n with exactly 2 local maxima.

Original entry on oeis.org

0, 3, 412, 50080, 6016512, 722051072, 86646800384, 10397622337536, 1247714738176000, 149725769101213696, 17967092296776155136, 2156051075653940805632, 258726129078829378961408, 31047135489462617851822080, 3725656258735540805375623168, 447078751048265125343493357568

Offset: 1

R. H. Hardin, Dec 06 2008

nonn

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

Cf. A152494, A334773.

a(n) = {(13*120^(n-1) - 13*8^(n-1) - 14*(n-1)*8^(n-1))/448} \\ Andrew Howroyd, May 10 2020

a(n) = (13*120^(n-1) - 13*8^(n-1) - 14*(n-1)*8^(n-1))/448. - Andrew Howroyd, May 10 2020

Terms a(7) and beyond from Andrew Howroyd, May 10 2020

A152518 1/36 of the number of permutations of 8 indistinguishable copies of 1..n with exactly 2 local maxima.

Original entry on oeis.org

0, 7, 1290, 214713, 35450244, 5849546139, 965177888238, 159254380788525, 26276973131433672, 4335700569742873071, 715390594038180275346, 118039448016603095674977, 19476508922742491987034060, 3213623972252540268102877443, 530247955421669426384081722614

Offset: 1

R. H. Hardin, Dec 06 2008

Andrew Howroyd, Table of n, a(n) for n = 1..200
Index entries for linear recurrences with constant coefficients, signature (183,-3051,13365).

Cf. A152494, A334773.

a(n) = {(97*165^(n-1) - 97*9^(n-1) - 104*(n-1)*9^(n-1))/2028} \\ Andrew Howroyd, May 10 2020

concat(0, Vec(x^2*(7 + 9*x) / ((1 - 9*x)^2*(1 - 165*x)) + O(x^17))) \\ Colin Barker, Jul 18 2020

a(n) = (97*165^(n-1) - 97*9^(n-1) - 104*(n-1)*9^(n-1))/2028. - Andrew Howroyd, May 10 2020
From Colin Barker, Jul 18 2020: (Start)
G.f.: x^2*(7 + 9*x) / ((1 - 9*x)^2*(1 - 165*x)).
a(n) = 183*a(n-1) - 3051*a(n-2) + 13365*a(n-3) for n>3.
(End)

Terms a(7) and beyond from Andrew Howroyd, May 10 2020

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A334774 Triangle read by rows: T(n,k) is the number of permutations of 2 indistinguishable copies of 1..n with exactly k local maxima.

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A152494 1/3 of the number of permutations of 2 indistinguishable copies of 1..n with exactly 2 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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A152499 1/12 of the number of permutations of 3 indistinguishable copies of 1..n with exactly 2 local maxima.

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A152504 1/10 of the number of permutations of 4 indistinguishable copies of 1..n with exactly 2 local maxima.

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A152509 1/30 of the number of permutations of 5 indistinguishable copies of 1..n with exactly 2 local maxima.

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Mathematica

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A152513 1/21 of the number of permutations of 6 indistinguishable copies of 1..n with exactly 2 local maxima.

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A152517 1/56 of the number of permutations of 7 indistinguishable copies of 1..n with exactly 2 local maxima.