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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 *) CROSSREFS Cf. A211318. Diagonals give: A001250, A001251, A001252, A001253, A230129, A230130, A230131, A230132, A230133. Sequence in context: A067228 A229756 A227450 * A059427 A137777 A126984 Adjacent sequences:  A010023 A010024 A010025 * A010027 A010028 A010029 KEYWORD nonn,tabl,nice AUTHOR EXTENSIONS Edited by N. J. A. Sloane, May 02 2012 STATUS approved

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