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A224290 Number of permutations of length n containing exactly 3 occurrences of 123 and 3 occurrences of 132. 1
0, 0, 0, 0, 0, 1, 6, 30, 136, 566, 2176, 7808, 26440, 85332, 264632, 793792, 2315136, 6592640, 18390784, 50392064, 135921664, 361536512, 949708800, 2466807808, 6342115328, 16153509888, 40790523904, 102186352640, 254105092096, 627533152256, 1539764125696 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,7

LINKS

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

B. Nakamura, Approaches for enumerating permutations with a prescribed number of occurrences of patterns, arXiv 1301.5080 [math.CO], 2013.

B. Nakamura, A Maple package for enumerating n-permutations with r occurrences of the pattern 123 and s occurrences of the pattern 132 [Broken link]

Index entries for linear recurrences with constant coefficients, signature (14,-84,280,-560,672,-448,128).

FORMULA

G.f.: -(4*x^8-8*x^7+24*x^6-36*x^5+62*x^4-60*x^3+30*x^2-8*x+1)*x^5 / (2*x-1)^7. - Alois P. Heinz, Apr 03 2013

From Colin Barker, Nov 28 2018: (Start)

a(n) = (1/9)*2^(n-15) * (307008 - 247512*n + 78118*n^2 - 12087*n^3 + 937*n^4 - 33*n^5 + n^6) for n>6.

a(n) = 14*a(n-1) - 84*a(n-2) + 280*a(n-3) - 560*a(n-4) + 672*a(n-5) - 448*a(n-6) + 128*a(n-7) for n>13.

(End)

EXAMPLE

a(5) = 1: (1,4,3,2,5).

a(6) = 6: (2,5,4,3,1,6), (2,5,4,3,6,1), (3,5,1,4,6,2), (3,6,1,4,2,5), (5,1,4,3,2,6), (6,1,4,3,2,5).

MAPLE

# Programs can be obtained from the Nakamura link

MATHEMATICA

Join[{0, 0, 0, 0, 0, 1, 6}, LinearRecurrence[{14, -84, 280, -560, 672, -448, 128}, {30, 136, 566, 2176, 7808, 26440, 85332}, 33]] (* Jean-Fran├žois Alcover, Nov 28 2018 *)

PROG

(PARI) concat([0, 0, 0, 0, 0], Vec(x^5*(1 - 8*x + 30*x^2 - 60*x^3 + 62*x^4 - 36*x^5 + 24*x^6 - 8*x^7 + 4*x^8) / (1 - 2*x)^7 + O(x^40))) \\ Colin Barker, Nov 28 2018

CROSSREFS

Cf. A000079, A001787, A001815, A046718, A001793.

Sequence in context: A317755 A036068 A162743 * A081895 A030280 A034545

Adjacent sequences:  A224287 A224288 A224289 * A224291 A224292 A224293

KEYWORD

nonn

AUTHOR

Brian Nakamura, Apr 03 2013

STATUS

approved

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Last modified May 22 05:03 EDT 2019. Contains 323473 sequences. (Running on oeis4.)