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 A010026 Triangle read by rows: number of permutations of 1..n by length of longest run. 16
 2, 2, 4, 2, 12, 10, 2, 16, 70, 32, 2, 20, 134, 442, 122, 2, 24, 198, 1164, 3108, 544, 2, 28, 274, 2048, 10982, 24216, 2770, 2, 32, 362, 3204, 22468, 112354, 208586, 15872, 2, 36, 462, 4720, 39420, 264538, 1245676, 1972904, 101042 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 2,1 REFERENCES F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 262. (Probably contains errors for n >= 13.) LINKS Alois P. Heinz, Rows n = 2..70, flattened EXAMPLE Triangle begins:   2,   2,  4,   2, 12,  10,   2, 16,  70,   32,   2, 20, 134,  442,   122,   2, 24, 198, 1164,  3108,    544,   2, 28, 274, 2048, 10982,  24216,   2770,   2, 32, 362, 3204, 22468, 112354, 208586, 15872, ... The row "2, 12, 10" for example means that there are two permutations of [1..4] in which the longest run up or down has length 4, 12 in which the longest run has length 3, and 10 in which the longest run has length 2. The following table, computed by Sean A. Irvine, May 02, 2012, gives an extended version of the triangle, oriented the right way round (cf. A211318), and corrects errors in David Kendall and Barton: n l=0, l=1, l=2, l=3, etc. ---------------------------- 1 [0, 1] 2 [0, 0, 2] 3 [0, 0, 4, 2] 4 [0, 0, 10, 12, 2] 5 [0, 0, 32, 70, 16, 2] 6 [0, 0, 122, 442, 134, 20, 2] 7 [0, 0, 544, 3108, 1164, 198, 24, 2] 8 [0, 0, 2770, 24216, 10982, 2048, 274, 28, 2] 9 [0, 0, 15872, 208586, 112354, 22468, 3204, 362, 32, 2] 10 [0, 0, 101042, 1972904, 1245676, 264538, 39420, 4720, 462, 36, 2] 11 [0, 0, 707584, 20373338, 14909340, 3340962, 514296, 64020, 6644, 574, 40, 2] 12 [0, 0, 5405530, 228346522, 191916532, 45173518, 7137818, 913440, 98472, 9024, 698, 44, 2] 13 [0, 0, 44736512, 2763212980, 2646100822, 652209564, 105318770, 13760472, 1523808, 145080, 11908, 834, 48, 2] 14 [0, 0, 398721962, 35926266244, 38932850396, 10024669626, 1649355338, 219040274, 24744720, 2419872, 206388, 15344, 982, 52, 2] 15 [0, 0, 3807514624, 499676669254, 609137502242, 163546399460, 27356466626, 3681354658, 422335056, 42129360, 3690960, 285180, 19380, 1142, 56, 2] MATHEMATICA (* This program is unsuited for a large number of terms *) f[p_List] := Max[Length /@ Split[Differences[p], #1*#2 > 0 &]] + 1; row[n_] := Sort[Tally[f /@ Permutations[Range[n]]], First[#1] > First[#2] &][[All, 2]]; Table[rn = row[n]; Print["n = ", n, " ", rn]; rn, {n, 2, 10}] // Flatten (* Jean-François Alcover, Mar 12 2014 *) T[n_, length_] := Module[{g, b}, g[u_, o_, t_] := g[u, o, t] = If[u+o == 0, 1, Sum[g[o + j - 1, u - j, 2], {j, 1, u}] + If[t

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Last modified June 20 13:10 EDT 2021. Contains 345164 sequences. (Running on oeis4.)