A230131 -id:A230131 - cp's OEIS frontend

A010026 Triangle read by rows: number of permutations of 1..n by length of longest run.

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

Offset: 2

N. J. A. Sloane

			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]A049293
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]

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.)

Alois P. Heinz, Rows n = 2..70, flattened

Cf. A211318. Diagonals give: A001250, A001251, A001252, A001253, A230129, A230130, A230131, A230132, A230133.

(* 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[tJean-François Alcover, Aug 18 2018, after Alois P. Heinz *)

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

A211318 Triangle read by rows: number of permutations of 1..n by length l of longest run (n >= 1, 1 <= l <= n).

Original entry on oeis.org

1, 0, 2, 0, 4, 2, 0, 10, 12, 2, 0, 32, 70, 16, 2, 0, 122, 442, 134, 20, 2, 0, 544, 3108, 1164, 198, 24, 2, 0, 2770, 24216, 10982, 2048, 274, 28, 2, 0, 15872, 208586, 112354, 22468, 3204, 362, 32, 2, 0, 101042, 1972904, 1245676, 264538, 39420, 4720, 462, 36, 2, 0

Offset: 1

N. J. A. Sloane, May 02 2012, based on computations by Sean A. Irvine

			Triangle begins:
n l=1, l=2, l=3, etc...
1 [1]
2 [0, 2]
3 [0, 4, 2]
4 [0, 10, 12, 2]
5 [0, 32, 70, 16, 2]
6 [0, 122, 442, 134, 20, 2]
7 [0, 544, 3108, 1164, 198, 24, 2]
8 [0, 2770, 24216, 10982, 2048, 274, 28, 2]
9 [0, 15872, 208586, 112354, 22468, 3204, 362, 32, 2]
10 [0, 101042, 1972904, 1245676, 264538, 39420, 4720, 462, 36, 2]
11 [0, 707584, 20373338, 14909340, 3340962, 514296, 64020, 6644, 574, 40, 2]
12 [0, 5405530, 228346522, 191916532, 45173518, 7137818, 913440, 98472, 9024, 698, 44, 2]
13 [0, 44736512, 2763212980, 2646100822, 652209564, 105318770, 13760472, 1523808, 145080, 11908, 834, 48, 2]
14 [0, 398721962, 35926266244, 38932850396, 10024669626, 1649355338, 219040274, 24744720, 2419872, 206388, 15344, 982, 52, 2]
15 [0, 3807514624, 499676669254, 609137502242, 163546399460, 27356466626, 3681354658, 422335056, 42129360, 3690960, 285180, 19380, 1142, 56, 2],
...
More rows than usual are shown, in order to correct errors in David, Kendall and Barton.

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 262. (Contains errors for n >= 13.)
Sean A. Irvine, Posting to Sequence Fans Mailing List, May 02 2012

Alois P. Heinz, Rows n = 1..70, flattened (rows n = 1..16 from Wouter Meeussen)
Max A. Alekseyev, On the number of permutations with bounded run lengths, arXiv:1205.4581 [math.CO], 2012-2013. [From _N. J. A. Sloane_, Oct 23 2012]

Mirror image of triangle in A010026.

Columns l=2-10 give: A001250, A001251, A001252, A001253, A230129, A230130, A230131, A230132, A230133.

<Wouter Meeussen, May 09 2012 *)
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[t1, 1] = 0;
Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Aug 18 2018, after Alois P. Heinz *)

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A010026 Triangle read by rows: number of permutations of 1..n by length of longest run.

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