A152499 -id:A152499 - cp's OEIS frontend

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.

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

Andrew Howroyd, May 10 2020

			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.

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

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

Cf. A334772, A334774, A334778.

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}

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.

A152500 1/4 the number of permutations of 3 indistinguishable copies of 1..n with exactly 3 local maxima.

Original entry on oeis.org

0, 1, 231, 21490, 1476084, 90050080, 5228286336, 297239712256, 16749407726592, 940343619493888, 52712719000338432, 2953100593082269696, 165399775808105742336, 9262957817232621568000, 518737995604927325405184, 29049593918675470746910720, 1626782962901824260072800256

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 (108,-3840,58752,-401664,1244160,-1433600).

Cf. A152499, A334774.

\\ PeaksBySig defined in A334774.
a(n) = {PeaksBySig(vector(n,i,3), [2])[1]/4} \\ Andrew Howroyd, May 12 2020

concat(0, Vec(x^2*(1 + 123*x + 382*x^2 - 16548*x^3 - 15440*x^4) / ((1 - 4*x)^3*(1 - 20*x)^2*(1 - 56*x)) + O(x^19))) \\ Colin Barker, Jul 19 2020

From Colin Barker, Jul 19 2020: (Start)
G.f.: x^2*(1 + 123*x + 382*x^2 - 16548*x^3 - 15440*x^4) / ((1 - 4*x)^3*(1 - 20*x)^2*(1 - 56*x)).
a(n) = 108*a(n-1) - 3840*a(n-2) + 58752*a(n-3) - 401664*a(n-4) + 1244160*a(n-5) - 1433600*a(n-6) for n>6.
(End)

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

A152501 1/8 the number of permutations of 3 indistinguishable copies of 1..n with exactly 4 local maxima.

Original entry on oeis.org

0, 0, 46, 22615, 5036741, 819235874, 114962084772, 14974498962192, 1876234090571968, 230313563301166336, 27966954502164518912, 3376705184454377873408, 406486565581361073979392, 48857132166440216820449280, 5867654791849010140880306176, 704409107074292841154786361344

Offset: 1

R. H. Hardin, Dec 06 2008

nonn

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

Cf. A152499, A334774.

\\ PeaksBySig defined in A334774.
a(n) = {PeaksBySig(vector(n,i,3), [3])[1]/8} \\ Andrew Howroyd, May 12 2020

Terms a(9) and beyond from Andrew Howroyd, May 12 2020

A152502 1/12 of the number of permutations of 3 indistinguishable copies of 1..n with exactly 5 local maxima.

Original entry on oeis.org

0, 0, 1, 7274, 6251162, 2764274116, 897380159188, 247392790837624, 62200280199674352, 14820288466400312448, 3420153590479988396800, 774303834249035054901248, 173288568985609322651099136, 38513999874946087671220207616, 8524401267844398602674455314432

Offset: 1

R. H. Hardin, Dec 06 2008

nonn

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

Cf. A152499, A334774.

\\ PeaksBySig defined in A334774.
a(n) = {PeaksBySig(vector(n,i,3), [4])[1]/12} \\ Andrew Howroyd, May 12 2020

Terms a(9) and beyond from Andrew Howroyd, May 12 2020

A152503 1/8 of the number of permutations of 3 indistinguishable copies of 1..n with exactly 6 local maxima.

Original entry on oeis.org

0, 0, 0, 925, 5134608, 7080780596, 5503883684118, 3175651343215500, 1543855504958661492, 676391857775294288488, 277604477433374392213008, 109265969423431070616562496, 41856404659462959845867172864, 15752452465692536904424614273024, 5860017184412283112523269770190848

Offset: 1

R. H. Hardin, Dec 06 2008

nonn

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

Cf. A152499, A334774.

\\ PeaksBySig defined in A334774.
a(n) = {PeaksBySig(vector(n,i,3), [5])[1]/8} \\ Andrew Howroyd, May 12 2020

Terms a(9) and beyond from Andrew Howroyd, May 12 2020

cp's OEIS Frontend

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.

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A152500 1/4 the number of permutations of 3 indistinguishable copies of 1..n with exactly 3 local maxima.

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A152501 1/8 the number of permutations of 3 indistinguishable copies of 1..n with exactly 4 local maxima.

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A152502 1/12 of the number of permutations of 3 indistinguishable copies of 1..n with exactly 5 local maxima.

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A152503 1/8 of the number of permutations of 3 indistinguishable copies of 1..n with exactly 6 local maxima.

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PARI

Extensions