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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-Fran├žois 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

N. J. A. Sloane

EXTENSIONS

Edited by N. J. A. Sloane, May 02 2012

STATUS

approved

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Last modified August 17 15:19 EDT 2017. Contains 290635 sequences.