A226435
Number of permutations of {1..n} with fewer than 2 interior elements having values lying between the values of their neighbors.
Original entry on oeis.org
1, 1, 2, 6, 22, 90, 422, 2226, 13102, 85170, 606542, 4697946, 39330982, 353985450, 3408792662, 34975509666, 380947661662, 4390028664930, 53368010874782, 682564606249386, 9162253729773142, 128794752680027610, 1892150024227428902, 28998220554100469106
Offset: 0
Some solutions for n=9:
..1...9...4...3...2...6...1...4...2...7...3...3...2...6...5....6
..7...2...7...1...5...3...7...1...5...2...8...1...3...1...4....3
..2...3...5...6...6...9...2...6...3...8...6...9...1...4...8....8
..4...1...6...4...3...1...5...5...1...6...9...4...5...3...1....1
..9...6...1...7...7...7...3...8...9...5...1...8...4...5...2....7
..5...5...9...9...4...5...9...3...6...9...5...5...9...7...9....4
..8...7...2...2...8...8...8...2...7...1...2...2...6...2...3....5
..3...4...8...8...1...4...4...9...4...4...7...7...8...9...7....9
..6...8...3...5...9...2...6...7...8...3...4...6...7...8...6....2
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CoefficientList[Series[Sec[x]+Tan[x] - (Sec[x]+Tan[x])^2 + (Sec[x]+Tan[x])^3, {x,0,20}], x] * Range[0,20]! (* Vaclav Kotesovec, Jun 11 2015 after Sergei N. Gladkovskii, all 210 terms match those in the b-file *)
{1}~Join~Table[Sum[(n - 2 i - 1) Sum[(-1)^(j + i)*2^(-n - j + 2 i + 2) StirlingS2[n, n + j - 2 i] Binomial[n + j - 2 i - 1, n - 2 i - 1] (n + j - 2 i)!, {j, 0, 2 i}], {i, 0, (n - 2)/2}], {n, 2, 22}] (* Michael De Vlieger, Apr 08 2016 *)
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a(n):=sum((n-2*i-1)*sum((-1)^(j+i)*2^(-n-j+2*i+2)*stirling2(n,n+j-2*i)*binomial(n+j-2*i-1,n-2*i-1)*(n+j-2*i)!,j,0,2*i),i,0,(n-2)/2); /* Vladimir Kruchinin, Apr 08 2016 */
A226436
Number of permutations of {1..n} with fewer than 3 interior elements having values lying between the values of their neighbors.
Original entry on oeis.org
1, 1, 2, 6, 24, 118, 658, 4078, 27724, 205134, 1641534, 14132390, 130299584, 1281352790, 13391026730, 148237230942, 1733059331604, 21341010960286, 276120222880246, 3745333514843734, 53149568086851784, 787599172764342822, 12166057777073762082
Offset: 0
Some solutions for n=9
..6....4....3....9....7....7....9....5....8....8....1....8....8....7....6....2
..1....2....2....2....3....4....3....3....1....4....4....3....4....1....4....1
..3....8....8....1....2....8....4....9....9....3....3....9....2....2....8....7
..9....3....7....6....8....3....2....1....4....7....5....5....9....6....5....9
..5....7....5....5....5....5....5....8....6....6....2....7....1....3....9....3
..2....9....6....3....1....9....7....2....7....2....8....4....5....5....2....6
..8....1....4....7....6....1....6....7....3....9....9....6....3....8....3....4
..4....5....9....4....4....6....1....4....5....1....7....2....6....4....1....5
..7....6....1....8....9....2....8....6....2....5....6....1....7....9....7....8
A226437
Number of permutations of {1..n} with fewer than 4 interior elements having values lying between the values of their neighbors.
Original entry on oeis.org
1, 1, 2, 6, 24, 120, 718, 4914, 37300, 308460, 2748354, 26194542, 265691456, 2856606480, 32449828310, 388371994890, 4885210675692, 64439603536980, 889564376280586, 12827733624005958, 192897097090449208, 3020051018689640760, 49155675298520617182, 830625328754347651746
Offset: 0
Some solutions for n=8:
..3....8....1....8....2....4....2....8....4....5....8....7....6....6....7....2
..7....1....2....5....7....3....5....4....7....7....3....1....3....8....3....8
..1....2....8....6....3....8....4....7....8....1....4....4....7....4....4....6
..2....4....6....3....6....7....8....5....3....2....6....3....5....7....5....3
..8....5....3....1....8....2....1....1....2....4....1....2....4....3....6....5
..6....3....5....2....4....5....3....2....6....3....2....5....8....1....1....1
..4....6....4....7....5....6....6....3....5....8....7....6....2....5....8....4
..5....7....7....4....1....1....7....6....1....6....5....8....1....2....2....7
A226438
Number of permutations of {1..n} with fewer than 5 interior elements having values lying between the values of their neighbors.
Original entry on oeis.org
1, 1, 2, 6, 24, 120, 720, 5038, 40066, 353556, 3399246, 35142546, 387129588, 4515261680, 55502974820, 716557285110, 9689420220510, 136922907564972, 2018121531850482, 30972340185946810, 494190208052390752, 8186609585709287112, 140620153367124829272
Offset: 0
Some solutions for n=7
..3....1....7....3....6....5....6....4....6....4....1....1....4....3....2....5
..4....5....6....2....3....4....5....5....2....1....6....2....7....6....4....6
..7....3....5....4....5....7....2....2....1....7....3....7....2....1....6....1
..2....2....2....6....1....6....3....6....4....5....5....5....1....7....5....7
..5....4....4....1....2....3....7....3....3....2....4....4....6....5....3....2
..6....7....3....5....7....1....1....7....5....6....7....3....3....2....7....3
..1....6....1....7....4....2....4....1....7....3....2....6....5....4....1....4
A226439
Number of permutations of {1..n} with fewer than 6 interior elements having values lying between the values of their neighbors.
Original entry on oeis.org
1, 1, 2, 6, 24, 120, 720, 5040, 40318, 362370, 3600306, 38963958, 453658380, 5626972260, 73801171100, 1017828597300, 14699233982370, 221584676368350, 3477976352489742, 56725919426072106, 959792584260031648, 16822545496476258000, 305055135390508456872
Offset: 0
Some solutions for n=7
..2....1....6....2....2....6....5....2....6....5....1....2....6....4....5....5
..1....2....3....7....3....5....3....5....3....6....7....7....1....7....1....4
..6....7....4....3....4....7....6....6....2....7....4....6....4....1....2....7
..5....4....2....5....7....2....4....4....5....1....3....3....3....2....7....2
..7....3....5....4....1....3....2....1....4....4....6....5....5....6....6....3
..4....5....1....1....5....4....1....3....1....3....2....4....2....5....4....6
..3....6....7....6....6....1....7....7....7....2....5....1....7....3....3....1
A226440
Number of permutations of {1..n} with fewer than 7 interior elements having values lying between the values of their neighbors.
Original entry on oeis.org
1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362878, 3627778, 39830282, 475089392, 6094561180, 83333937940, 1205398303180, 18331873363480, 291706615484450, 4838414115229430, 83404373736108502, 1490681666580459304, 27572241018065482032, 526959942343051879032
Offset: 0
Some solutions for n=7
..2....2....2....1....7....2....6....4....2....2....7....1....1....4....6....4
..5....3....1....7....4....3....7....2....1....7....3....3....5....3....3....5
..3....7....6....2....5....6....2....1....5....6....5....4....4....2....2....6
..4....1....4....4....2....7....1....5....6....4....6....6....7....1....4....2
..7....4....3....3....3....4....4....3....4....5....2....7....3....5....7....3
..1....6....5....5....6....1....5....6....3....3....1....2....2....6....1....7
..6....5....7....6....1....5....3....7....7....1....4....5....6....7....5....1
Showing 1-6 of 6 results.