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<length, Sum[g[u + j - 1, o - j, t+1], {j, 1, o}], 0]];
b[u_, o_, t_] := b[u, o, t] = If[t == length, g[u, o, t], Sum[b[o + j - 1, u - j, 2], {j, 1, u}] + Sum[b[u + j - 1, o - j, t + 1], {j, 1, o}]]; Sum[b[j - 1, n - j, 1], {j, 1, n}]];
Table[T[n, n-lg+1], {n, 2, 10}, {lg, 1, n-1}] // Flatten (* Jean-François Alcover, Aug 18 2018, after Alois P. Heinz *)
CROSSREFS
KEYWORD
AUTHOR
EXTENSIONS
Edited by N. J. A. Sloane, May 02 2012
STATUS
approved