cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 18 results. Next

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.

Original entry on oeis.org

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

Views

Author

Andrew Howroyd, May 11 2020

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

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.

Programs

  • PARI
    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))) }

Formula

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).

A334773 Array read by antidiagonals: T(n,k) is the number of permutations of k indistinguishable copies of 1..n with exactly 2 local maxima.

Original entry on oeis.org

3, 12, 57, 30, 360, 705, 60, 1400, 7968, 7617, 105, 4170, 51750, 163584, 78357, 168, 10437, 241080, 1830000, 3293184, 791589, 252, 23072, 894201, 13562040, 64168750, 65968128, 7944321, 360, 46440, 2804480, 75278553, 759940800, 2246625000, 1319854080, 79541625
Offset: 2

Views

Author

Andrew Howroyd, May 10 2020

Keywords

Examples

			Array begins:
======================================================
n\k |       2          3           4              5
----|-------------------------------------------------
  2 |       3         12          30            60 ...
  3 |      57        360        1400          4170 ...
  4 |     705       7968       51750        241080 ...
  5 |    7617     163584     1830000      13562040 ...
  6 |   78357    3293184    64168750     759940800 ...
  7 |  791589   65968128  2246625000   42560067360 ...
  8 | 7944321 1319854080 78636093750 2383387566720 ...
  ...
The T(2,2) = 3 permutations of 1122 with 2 local maxima are 1212, 2112, 2121.

Links

Crossrefs

Columns k=2..8 are 3*A152494, 12*A152499, 10*A152504, 30*A152509, 21*A152513, 56*A152517, 36*A152518.
Cf. A334772, A334774, A334778.

Programs

  • PARI
    T(n,k) = {3*((k^2 + 4*k + 1)*binomial(k+3,3)^(n-1) - (2*k^2 + 9*k + 1)*(k+1)^(n-1) - k*(k + 5)*(n-2)*(k+1)^(n-1))/(k + 5)^2}

Formula

T(n,k) = Sum_{j=0..n-2} P(k-1,3) * P(k-2,2) * P(k,2)^(n-2-j) * P(k,4)^j + 2 * (n-j-2) * P(k-1,3)^2 * P(k,2)^(n-3-j) * P(k,4)^j where P(n,k) = binomial(n+k-1,k-1).
T(n,k) = 3*((k^2 + 4*k + 1)*binomial(k+3,3)^(n-1) - (2*k^2 + 9*k + 1)*(k+1)^(n-1) - k*(k + 5)*(n-2)*(k+1)^(n-1))/(k + 5)^2.

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

Views

Author

Andrew Howroyd, May 10 2020

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

Columns k=2..6 are A159716, A159722, A159728, A159734, A159737.
Cf. A334773, A334774, A334778.

Programs

  • PARI
    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)}

Formula

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).

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

Views

Author

R. H. Hardin, Apr 20 2009

Keywords

Links

Crossrefs

Column 3 of A334778.
Cf. A159716.

Programs

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

Extensions

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

Views

Author

R. H. Hardin, Apr 20 2009

Keywords

Links

Crossrefs

Column 4 of A334778.
Cf. A159716.

Programs

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

Extensions

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

Views

Author

R. H. Hardin, Apr 20 2009

Keywords

Links

Crossrefs

Column 5 of A334778.
Cf. A159716.

Programs

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

Extensions

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

Views

Author

R. H. Hardin, Apr 20 2009

Keywords

Links

Crossrefs

Column 6 of A334778.
Cf. A159716.

Programs

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

Extensions

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

A334779 Number of permutations of 2 indistinguishable copies of 1..n arranged in a circle with exactly n local maxima.

Original entry on oeis.org

1, 1, 2, 6, 88, 1480, 40272, 1476944, 71865728, 4454840448, 343059301120, 32120367997696, 3593423957597184, 473387964389590016, 72533147097036867584, 12789470502905585018880, 2571336187676016073080832, 584713932920281483718459392, 149314675071137196105777217536
Offset: 0

Views

Author

Andrew Howroyd, May 13 2020

Keywords

Links

Crossrefs

Main diagonal of A334778.

Programs

  • PARI
    \\ CircPeaksBySig defined in A334778.
a(n) = {if(n==0, 1, CircPeaksBySig(vector(n, i, 2), [n])[1])} \\ Andrew Howroyd, May 13 2020

A334780 Total number of local maxima in all permutations of 2 indistinguishable copies of 1..n arranged in a circle.

Original entry on oeis.org

1, 8, 168, 6336, 360000, 28728000, 3067243200, 422479411200, 72968229734400, 15446903472000000, 3933958530902400000, 1186723982675911680000, 418497083472072084480000, 170595438771094235458560000, 79604772050991371184000000000, 42160341403338557272473600000000
Offset: 1

Views

Author

Andrew Howroyd, May 13 2020

Keywords

Links

Crossrefs

Cf. A334778.

Programs

  • PARI
    \\ CircPeaksBySig defined in A334778.
a(n)={my(u=CircPeaksBySig(vector(n, i, 2), [1..n])); sum(k=1, #u, k*u[k])}

Formula

a(n) = Sum_{k=1..n} k*A334778(n,k).

A159723 Number of permutations of 3 indistinguishable copies of 1..n arranged in a circle with exactly 3 local maxima.

Original entry on oeis.org

0, 2, 924, 130672, 11798800, 882163680, 60299781248, 3933244436480, 249755233812480, 15590436752711680, 961617113393102848, 58776390094965178368, 3566492692755774439424, 215104023517101044006912, 12906638765341805641728000, 770965701556822810254376960
Offset: 1

Views

Author

R. H. Hardin, Apr 20 2009

Keywords

Links

Crossrefs

Cf. A159721, A159722, A334778.

Programs

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

Extensions

Terms a(9) and beyond from Andrew Howroyd, May 13 2020
Showing 1-10 of 18 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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