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A104002 Triangle T(n,k) read by rows: number of permutations in S_n avoiding all k-length patterns that start with 1 except one fixed pattern and containing it exactly once. 3
1, 2, 1, 3, 4, 1, 4, 12, 6, 1, 5, 32, 27, 8, 1, 6, 80, 108, 48, 10, 1, 7, 192, 405, 256, 75, 12, 1, 8, 448, 1458, 1280, 500, 108, 14, 1, 9, 1024, 5103, 6144, 3125, 864, 147, 16, 1, 10, 2304, 17496, 28672, 18750, 6480, 1372, 192, 18, 1, 11, 5120, 59049, 131072 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

2,2

COMMENTS

T(n+k,k+1) = total number of occurrences of any given letter in all possible n-length words on a k-letter alphabet. For example, with the 2 letter alphabet {0,1} there are 4 possible 2-length words: {00,01,10,11}. The letter 0 occurs 4 times altogether, as does the letter 1. T(4,3) = 4. - Ross La Haye, Jan 03 2007

Table T(n,k) = k*n^(k-1) n,k > 0 read by antidiagonals. - Boris Putievskiy, Dec 17 2012

LINKS

Michael De Vlieger, Table of n, a(n) for n = 2..11176 (rows 2 <= n <= 150).

T. Mansour, Permutations containing and avoiding certain patterns, arXiv:math/9911243 [math.CO], 1999-2000.

Boris Putievskiy, Transformations Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.

Franck Ramaharo, A generating polynomial for the pretzel knot, arXiv:1805.10680 [math.CO], 2018.

FORMULA

T(n, k) = (n-k+1) * (k-1)^(n-k), k<=n.

As a linear array, the sequence is a(n) = A004736(n)*A002260(n)^(A004736(n)-1) or a(n) = ((t*t+3*t+4)/2-n)*(n-(t*(t+1)/2))^((t*t+3*t+4)/2-n-1), where t=floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Dec 17 2012

EXAMPLE

Triangle begins:

  1;

  2,   1;

  3,   4,    1;

  4,  12,    6,    1;

  5,  32,   27,    8,   1;

  6,  80,  108,   48,  10,   1;

  7, 192,  405,  256,  75,  12,  1;

  8, 448, 1458, 1280, 500, 108, 14, 1;

MATHEMATICA

Table[(n - k + 1) (k - 1)^(n - k), {n, 2, 12}, {k, 2, n}] // Flatten (* Michael De Vlieger, Aug 22 2018 *)

CROSSREFS

Cf. Left-edge columns include A001787, A027471, A002697, A053464, A053469, A027473, A053539, A053540, A053541, A081127, A081128.

Sequence in context: A137649 A180915 A240783 * A073135 A063804 A213800

Adjacent sequences:  A103999 A104000 A104001 * A104003 A104004 A104005

KEYWORD

nonn,tabl

AUTHOR

Ralf Stephan, Feb 26 2005

STATUS

approved

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Last modified March 21 22:19 EDT 2019. Contains 321382 sequences. (Running on oeis4.)