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Number T(n,k) of words on {1,1,2,2,...,n,n} with longest increasing subsequence of length k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
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%I #19 May 09 2018 09:51:00

%S 1,0,1,0,1,5,0,1,42,47,0,1,351,1527,641,0,1,3113,43910,54987,11389,0,

%T 1,29003,1302660,3844840,2059147,248749,0,1,280220,40970298,265777225,

%U 285588543,82025038,6439075,0,1,2782475,1364750889,19104601915,37783672691,19773928713,3507289363,192621953

%N Number T(n,k) of words on {1,1,2,2,...,n,n} with longest increasing subsequence of length k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

%H Alois P. Heinz, <a href="/A267480/b267480.txt">Rows n = 0..18, flattened</a>

%H Ferenc Balogh, <a href="https://arxiv.org/abs/1505.01389">A generalization of Gessel's generating function to enumerate words with double or triple occurrences in each letter and without increasing subsequences of a given length</a>, arXiv:1505.01389, 2015

%H Shalosh B. Ekhad and Doron Zeilberger, <a href="http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/sloane75.html">The Generating Functions Enumerating 12..d-Avoiding Words with r occurrences of each of 1,2, ..., n are D-finite for all d and all r</a>, 2014

%F T(n,0) = A267479(n,0), T(n,k) = A267479(n,k) - A267479(n,k-1) for k>0.

%F Sum_{k=0..n-1} T(n,k) = A267532(n).

%e Triangle T(n,k) begins:

%e 1;

%e 0, 1;

%e 0, 1, 5;

%e 0, 1, 42, 47;

%e 0, 1, 351, 1527, 641;

%e 0, 1, 3113, 43910, 54987, 11389;

%e 0, 1, 29003, 1302660, 3844840, 2059147, 248749;

%e 0, 1, 280220, 40970298, 265777225, 285588543, 82025038, 6439075;

%Y Main diagonal gives A006902.

%Y Row sums give A000680.

%Y Cf. A267479, A267532.

%K nonn,tabl

%O 0,6

%A _Alois P. Heinz_, Jan 15 2016