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Number a(n,k) of positions (cyclic permutations) of circular permutations of [n] with exactly k (unspecified) increasing or decreasing modular runs (3-sequences), with clockwise and counterclockwise traversals counted as distinct; triangle a(n,k) read by rows, 0<=k<=n.
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%I #26 Jan 24 2014 16:11:41

%S 0,0,0,0,0,0,0,0,0,6,16,0,0,0,8,60,50,0,0,0,10,456,144,108,0,0,0,12,

%T 3458,1078,294,196,0,0,0,14,29296,7936,2240,512,320,0,0,0,16,275166,

%U 66096,16200,4104,810,486,0,0,0,18,2843980,611200,135600,29200,6900,1200,700,0,0,0,20

%N Number a(n,k) of positions (cyclic permutations) of circular permutations of [n] with exactly k (unspecified) increasing or decreasing modular runs (3-sequences), with clockwise and counterclockwise traversals counted as distinct; triangle a(n,k) read by rows, 0<=k<=n.

%C Arrangements that differ in the direction in which the cycle is traversed count as different.

%D Paul J. Campbell, Circular permutations with exactly one modular run (3-sequence), submitted to Journal of Integer Sequences

%H Paul J. Campbell, <a href="/A235943/b235943.txt">Rows n = 0..13, flattened</a>

%H Paul J. Campbell, <a href="/A235943/a235943_2.txt">Table of rows n = 0..13 of A235943</a>

%H Wayne M. Dymáček and Isaac Lambert, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL14/Dymacek/dymacek5.html">Circular permutations avoiding runs of i, i+1, i+2 or i, i-1, i-2</a>, Journal of Integer Sequences, Vol. 14 (2011) Article 11.1.6.

%Y Cf. A165961, A165964, A165962, A078628, A078673.

%Y Cf. A235937, A235938, A235939, A235940, A235941, A235942.

%K nonn,tabl

%O 0,10

%A _Paul J. Campbell_, Jan 20 2014, with Joe Marasco and Ashish Vikram