A152509 -id:A152509

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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.

+10
10

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

2,1

LINKS

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

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.

EXAMPLE

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.

PROG

(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}

CROSSREFS

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

Cf. A334772, A334774, A334778.

KEYWORD

nonn,tabl

AUTHOR

Andrew Howroyd, May 10 2020

STATUS

approved

A152510

1/60 of the number of permutations of 5 indistinguishable copies of 1..n with exactly 3 local maxima.

+10
1

0, 2, 1066, 328314, 87554515, 22414176982, 5672480870616, 1431066048773744, 360732335571459920, 90911141639422741152, 22910020941551289849856, 5773350885207751422091264, 1454885995214232796339050240, 366631366567387199476086758912, 92391110171365499708617443239936

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

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

Index entries for linear recurrences with constant coefficients, signature (382,-38020,1394280,-17690400,92123136,-170698752).

FORMULA

From Colin Barker, Jul 19 2020: (Start)

G.f.: x^2*(2 + 302*x - 2858*x^2 - 120673*x^3 - 71148*x^4) / ((1 - 6*x)^3*(1 - 56*x)^2*(1 - 252*x)).

a(n) = 382*a(n-1) - 38020*a(n-2) + 1394280*a(n-3) - 17690400*a(n-4) + 92123136*a(n-5) - 170698752*a(n-6) for n>6.

(End)

MATHEMATICA

LinearRecurrence[{382, -38020, 1394280, -17690400, 92123136, -170698752}, {0, 2, 1066, 328314, 87554515, 22414176982}, 20] (* Harvey P. Dale, Mar 14 2022 *)

PROG

(PARI) \\ PeaksBySig defined in A334774.

a(n) = {PeaksBySig(vector(n, i, 5), [2])[1]/60} \\ Andrew Howroyd, May 12 2020

(PARI) concat(0, Vec(x^2*(2 + 302*x - 2858*x^2 - 120673*x^3 - 71148*x^4) / ((1 - 6*x)^3*(1 - 56*x)^2*(1 - 252*x)) + O(x^20))) \\ Colin Barker, Jul 19 2020

CROSSREFS

Cf. A152509, A334774.

KEYWORD

nonn,easy

AUTHOR

R. H. Hardin, Dec 06 2008

EXTENSIONS

Terms a(7) and beyond from Andrew Howroyd, May 12 2020

STATUS

approved

A152511

1/60 of the number of permutations of 5 indistinguishable copies of 1..n with exactly 4 local maxima.

+10
1

0, 1, 4194, 5825786, 5682784528, 4873147413516, 3978083870212150, 3186605615943562016, 2534375865966184697328, 2010275266425805924583168, 1593002777198909770195152928, 1261900375041824878511515546368, 999492236937050258502770760302080, 791616311022376735886612880860890112

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

LINKS

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

PROG

(PARI) \\ PeaksBySig defined in A334774.

a(n) = {PeaksBySig(vector(n, i, 5), [3])[1]/60} \\ Andrew Howroyd, May 12 2020

CROSSREFS

Cf. A152509, A334774.

KEYWORD

nonn

AUTHOR

R. H. Hardin, Dec 06 2008

EXTENSIONS

Terms a(7) and beyond from Andrew Howroyd, May 12 2020

STATUS

approved

A152512

1/6 of the number of permutations of 5 indistinguishable copies of 1..n with exactly 5 local maxima.

+10
1

0, 1, 51408, 331072352, 1080698915350, 2691500727775616, 5953257961411738328, 12474940206857421730672, 25498614004537897031551640, 51527434528518884637847012176, 103587934174554666918594336695328, 207763415020909344351469183584249792

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

LINKS

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

PROG

(PARI) \\ PeaksBySig defined in A334774.

a(n) = {PeaksBySig(vector(n, i, 5), [4])[1]/6} \\ Andrew Howroyd, May 12 2020

CROSSREFS

Cf. A152509, A334774.

KEYWORD

nonn

AUTHOR

R. H. Hardin, Dec 06 2008

EXTENSIONS

Terms a(7) and beyond from Andrew Howroyd, May 12 2020

STATUS

approved

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