The On-Line Encyclopedia of Integer Sequences (OEIS)

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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.
(history; published version)

#23 by Michel Marcus at Tue Feb 15 12:57:15 EST 2022

STATUS

editing

approved

#22 by Michel Marcus at Tue Feb 15 12:57:08 EST 2022

LINKS

Boris Putievskiy, <a href="http://arxiv.org/abs/1212.2732">Transformations Integer Sequences And Pairing Functions</a> , arXiv:1212.2732 [math.CO], 2012.

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approved

editing

Discussion

Tue Feb 15

12:57

Michel Marcus: just a comma

#21 by Susanna Cuyler at Thu Aug 23 02:18:53 EDT 2018

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proposed

approved

#20 by Michel Marcus at Wed Aug 22 10:52:40 EDT 2018

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editing

proposed

#19 by Michel Marcus at Wed Aug 22 10:52:33 EDT 2018

LINKS

T. Mansour, <a href="httphttps://arXivarxiv.org/abs/math.CO/9911243">Permutations containing and avoiding certain patterns</a>, arXiv:math/9911243 [math.CO], 1999-2000.

Boris Putievskiy, <a href="http://arxiv.org/abs/1212.2732">Transformations Integer Sequences And Pairing Functions</a> arXiv:1212.2732 [math.CO], 2012.

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proposed

editing

#18 by Michael De Vlieger at Wed Aug 22 10:41:08 EDT 2018

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editing

proposed

#17 by Michael De Vlieger at Wed Aug 22 10:41:02 EDT 2018

LINKS

Michael De Vlieger, <a href="/A104002/b104002.txt">Table of n, a(n) for n = 2..11176</a> (rows 2 <= n <= 150).

Franck Ramaharo, <a href="https://arxiv.org/abs/1805.10680">A generating polynomial for the pretzel knot</a>, arXiv:1805.10680 [math.CO], 2018.

MATHEMATICA

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

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approved

editing

#16 by N. J. A. Sloane at Sat Jun 24 00:55:54 EDT 2017

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proposed

approved

#15 by Jon E. Schoenfield at Fri Jun 23 23:47:55 EDT 2017

STATUS

editing

proposed

#14 by Jon E. Schoenfield at Fri Jun 23 23:47:51 EDT 2017

COMMENTS

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

FORMULA

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;

STATUS

approved

editing