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 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. 16
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Sum_{k=0..1} T(n,k) = A088218(n). Sum_{k=0..2} T(n,k) = A239295(n). Sum_{k=0..3} T(n,k) = A239299(n). Sum_{k=1..n} k * T(n,k) = A275576(n). LINKS Alois P. Heinz, Rows n = 0..18, flattened EXAMPLE 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;   ... MAPLE 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); MATHEMATICA 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]]0, A268869, A268870, A268871, A268872, A268873, A268874, A268875, A268876, A268877. Main diagonal gives A000012. T(n,n-1) gives A152618(n) for n>0. T(n,n-2) gives A268936(n). T(2n,n) gives A268949(n). Row sums give A000312. Cf. A047874, A239295, A239299, A275576. Sequence in context: A202995 A191578 A288385 * A067176 A249480 A271704 Adjacent sequences:  A245664 A245665 A245666 * A245668 A245669 A245670 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Jul 28 2014 STATUS approved

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Last modified January 21 12:13 EST 2022. Contains 350477 sequences. (Running on oeis4.)