The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A008970

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A008970

Triangle T(n,k) = P(n,k)/2, n >= 2, 1 <= k < n, of one-half of number of permutations of 1..n such that the differences have k runs with the same signs.
(history; published version)

#38 by Alois P. Heinz at Thu Feb 09 09:50:34 EST 2023

STATUS

editing

approved

#37 by Alois P. Heinz at Thu Feb 09 09:50:29 EST 2023

CROSSREFS

T(2n,n) gives A360426.

STATUS

approved

editing

#36 by Alois P. Heinz at Wed Feb 08 18:42:09 EST 2023

STATUS

editing

approved

#35 by Alois P. Heinz at Wed Feb 08 18:42:06 EST 2023

MAPLE

T:= proc(n, k) option remember; `if`(n<2, 0, `if`(k=1, 1,

k*T(n-1, k) + 2*T(n-1, k-1) + (n-k)*T(n-1, k-2)))

end:

seq(seq(T(n, k), k=1..n-1), n=2..12); # Alois P. Heinz, Feb 08 2023

#34 by Alois P. Heinz at Wed Feb 08 18:41:34 EST 2023

DATA

1, 1, 2, 1, 6, 5, 1, 14, 29, 16, 1, 30, 118, 150, 61, 1, 62, 418, 926, 841, 272, 1, 126, 1383, 4788, 7311, 5166, 1385, 1, 254, 4407, 22548, 51663, 59982, 34649, 7936, 1, 510, 13736, 100530, 325446, 553410, 517496, 252750, 50521, 1, 1022, 42236, 433162, 1910706, 4474002, 6031076, 4717222, 1995181, 353792

STATUS

approved

editing

#33 by Peter Luschny at Thu Aug 08 03:34:37 EDT 2019

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reviewed

approved

#32 by Joerg Arndt at Thu Aug 08 02:00:36 EDT 2019

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proposed

reviewed

#31 by Michel Marcus at Thu Aug 08 00:56:07 EDT 2019

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editing

proposed

#30 by Michel Marcus at Thu Aug 08 00:56:01 EDT 2019

LINKS

Désiré André, <a href="httphttps://www.numdamdoi.org/numdam-bin10.24033/item?id=ASENS_1884_3_1__121_0asens.235">Etude sur les maxima, minima et sequences des permutations</a>, Ann. Sci. Ecole Norm. Sup., 3, no. 1 (1884), 121-135.

M. Bona and R. Ehrenborg, <a href="httphttps://arXivarxiv.org/abs/math.CO/9902020">A combinatorial proof of the log-concavity of the numbers of permutations with k runs</a>, arXiv:math/9902020 [math.CO], 1999.

STATUS

proposed

editing

#29 by Jon E. Schoenfield at Thu Aug 08 00:50:09 EDT 2019

STATUS

editing

proposed