%I #9 Jul 17 2024 19:11:28

%S 1,1,2,1,2,4,1,2,6,10,1,2,6,22,32,1,2,6,24,90,122,1,2,6,24,118,422,

%T 544,1,2,6,24,120,658,2226,2770,1,2,6,24,120,718,4078,13102,15872,1,2,

%U 6,24,120,720,4914,27724,85170,101042,1,2,6,24,120,720,5038,37300,205134,606542,707584

%N T(n,k) = number of permutations of {1..n} with fewer than k interior elements having values lying between the values of their neighbors.

%C Table starts

%C .......1........1.........1.........1.........1.........1.........1.........1

%C .......2........2.........2.........2.........2.........2.........2.........2

%C .......4........6.........6.........6.........6.........6.........6.........6

%C ......10.......22........24........24........24........24........24........24

%C ......32.......90.......118.......120.......120.......120.......120.......120

%C .....122......422.......658.......718.......720.......720.......720.......720

%C .....544.....2226......4078......4914......5038......5040......5040......5040

%C ....2770....13102.....27724.....37300.....40066.....40318.....40320.....40320

%C ...15872....85170....205134....308460....353556....362370....362878....362880

%C ..101042...606542...1641534...2748354...3399246...3600306...3627778...3628798

%C ..707584..4697946..14132390..26194542..35142546..38963958..39830282..39914754

%C .5405530.39330982.130299584.265691456.387129588.453658380.475089392.478739984

%H R. H. Hardin, <a href="/A226441/b226441.txt">Table of n, a(n) for n = 1..9999</a>

%e Some solutions for n=8 k=4

%e ..5....8....1....7....5....5....8....4....7....5....3....4....7....6....5....1

%e ..8....1....7....3....1....4....1....7....2....4....5....2....2....5....3....2

%e ..6....5....2....6....3....8....7....5....1....3....7....1....5....4....7....7

%e ..7....6....6....8....6....6....6....1....5....8....2....7....1....7....6....4

%e ..3....7....3....2....7....3....3....3....6....1....6....5....8....1....1....3

%e ..4....3....8....1....4....7....2....8....4....7....4....8....3....8....2....6

%e ..1....4....5....5....8....2....5....6....3....6....1....6....4....3....4....8

%e ..2....2....4....4....2....1....4....2....8....2....8....3....6....2....8....5

%Y Column 1 is A001250.

%Y Cf. A008970, A059427.

%K nonn,tabl

%O 1,3

%A _R. H. Hardin_, Jun 06 2013