A245667
Number T(n,k) of sequences in {1,...,n}^n with longest increasing subsequence of length k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
Original entry on oeis.org
1, 0, 1, 0, 3, 1, 0, 10, 16, 1, 0, 35, 175, 45, 1, 0, 126, 1771, 1131, 96, 1, 0, 462, 17906, 23611, 4501, 175, 1, 0, 1716, 184920, 461154, 161876, 13588, 288, 1, 0, 6435, 1958979, 8837823, 5179791, 759501, 34245, 441, 1, 0, 24310, 21253375, 169844455, 157279903, 36156355, 2785525, 75925, 640, 1
Offset: 0
T(3,1) = 10: [1,1,1], [2,1,1], [2,2,1], [2,2,2], [3,1,1], [3,2,1], [3,2,2], [3,3,1], [3,3,2], [3,3,3].
T(3,3) = 1: [1,2,3].
Triangle T(n,k) begins:
1;
0, 1;
0, 3, 1;
0, 10, 16, 1;
0, 35, 175, 45, 1;
0, 126, 1771, 1131, 96, 1;
0, 462, 17906, 23611, 4501, 175, 1;
0, 1716, 184920, 461154, 161876, 13588, 288, 1;
...
Columns k=0-10 give:
A000007,
A088218 or
A001700(n-1) for n>0,
A268869,
A268870,
A268871,
A268872,
A268873,
A268874,
A268875,
A268876,
A268877.
-
b:= proc(n, l) option remember; `if`(n=0, 1, add(b(n-1, [seq(min(l[j],
`if`(j=1 or l[j-1] `if`(k=0, `if`(n=0, 1, 0), b(n, [n$k])):
T:= (n, k)-> A(n, k) -`if`(k=0, 0, A(n, k-1)):
seq(seq(T(n, k), k=0..n), n=0..9);
-
b[n_, l_List] := b[n, l] = If[n == 0, 1, Sum[b[n-1, Table[Min[l[[j]], If[j == 1 || l[[j-1]]Jean-François Alcover, Feb 04 2015, after Alois P. Heinz *)
A239368
Number of words of length n over the alphabet {0,...,n-1} that avoid the pattern 1111.
Original entry on oeis.org
1, 1, 4, 27, 252, 3020, 44220, 765030, 15269520, 345376080, 8730489600, 243911883600, 7463164262400, 248207881521600, 8915064168410400, 343923449355486000, 14182674669779616000, 622591172035376544000, 28986699477880400256000, 1426677017904959524704000
Offset: 0
-
a:= proc(n) option remember; `if`(n<3, n^n,
((105*n^3-252*n^2+175*n-36) *a(n-1) -2*(n-1)^2 *a(n-2)
+2*(5*n-2)*(n-1)^2*(n-2)^2*a(n-3)) / (4*(2*n-1)*(5*n-7)))
end:
seq(a(n), n=0..20); # Alois P. Heinz, Jul 20 2014
A268869
Number of sequences in {1,...,n}^n with longest increasing subsequence of length two.
Original entry on oeis.org
1, 16, 175, 1771, 17906, 184920, 1958979, 21253375, 235401166, 2653427140, 30356604642, 351714783980, 4119532070980, 48708027030608, 580682498991627, 6973355148949335, 84286653134676230, 1024694093358751200, 12522664845237058050, 153762682169941498170
Offset: 2
a(2) = 1: 12.
a(3) = 16: 112, 113, 121, 122, 131, 132, 133, 212, 213, 223, 231, 232, 233, 312, 313, 323.
a(4) = 175: 1112, 1113, 1114, 1121, ..., 4414, 4423, 4424, 4434.
A268870
Number of sequences in {1,...,n}^n with longest increasing subsequence of length three.
Original entry on oeis.org
1, 45, 1131, 23611, 461154, 8837823, 169844455, 3307239364, 65559881608, 1325285983528, 27321430823064, 573932303529170, 12269791047231748, 266575464412874571, 5877527995117635663, 131334452330324176828, 2970563969803965041900, 67935408457473145627500
Offset: 3
a(3) = 1: 123.
a(4) = 45: 1123, 1124, 1134, 1213, 1214, 1223, 1224, 1231, 1232, 1233, 1241, 1242, 1243, 1244, 1314, 1323, 1324, 1334, 1341, 1342, 1343, 1344, 1423, 1424, 1434, 2123, 2124, 2134, 2234, 2314, 2324, 2334, 2341, 2342, 2343, 2344, 2434, 3123, 3124, 3134, 3234, 4123, 4124, 4134, 4234.
A239296
Words of length n over the alphabet {0,...,n-1} that are 112-avoiding.
Original entry on oeis.org
1, 1, 4, 24, 186, 1745, 19090, 237594, 3305610, 50736447, 850285888, 15430858102, 301188960996, 6286897888336, 139661748755464, 3288136274970026, 81747690289414282, 2139280904017185007, 58762203796595526676, 1689933802493155288876, 50768897450201657287066
Offset: 0
For a(3) all words of length 3 over {0,1,2} except: 001, 002, 112.
A239371
Words of length n over the alphabet {0,...,n-1} that avoid the pattern 1123.
Original entry on oeis.org
1, 1, 4, 27, 252, 2935, 40351, 633864, 11139840, 215800335, 4557031407, 103989209082, 2546369980589, 66521801043100, 1845008781481516, 54102505102771686, 1671315770146956704, 54219694639758864087, 1842085959295755494339, 65380444351870047159358
Offset: 0
A239369
Words of length n over the alphabet {0,...,n-1} that avoid the pattern 1112.
Original entry on oeis.org
1, 1, 4, 27, 250, 2935, 41676, 693798, 13242762, 285008223, 6826605520, 180084705031, 5187350461800, 161993077533092, 5451141889485852, 196629624309733488, 7568494925621640330, 309628452894632990415, 13415755128012445232100
Offset: 0
A239370
Words of length n over the alphabet {0,...,n-1} that avoid the pattern 1122.
Original entry on oeis.org
1, 1, 4, 27, 250, 2915, 40806, 664944
Offset: 0
A239372
Words of length n over the alphabet {0,...,n-1} that avoid the pattern 1212.
Original entry on oeis.org
1, 1, 4, 27, 250, 2915, 40746, 661304
Offset: 0
A239373
Words of length n over the alphabet {0,...,n-1} that avoid the pattern 1213.
Original entry on oeis.org
1, 1, 4, 27, 252, 2935, 40330, 632408
Offset: 0
Showing 1-10 of 16 results.
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