A390670 - OEIS

%I #17 Dec 01 2025 05:56:47

%S 1,0,1,0,1,1,0,2,4,1,0,9,15,9,1,0,80,79,64,16,1,0,1226,705,471,185,25,

%T 1,0,29215,10533,4286,1991,426,36,1,0,1004864,244406,56173,22446,6389,

%U 847,49,1,0,47241632,8257368,1134186,294934,93452,16903,1520,64,1

%N Triangle read by rows: T(n,k) is the number of irreducible words covering the alphabet [n] such that the maximal cardinality of C is k, where C is a subset of the alphabet such that all letters in C appear in weakly increasing order within the word.

%C Here a word is irreducible if no letters can be removed without changing the maximal cardinality of C. The lengths of these words range from n to (n-1)*2.

%H Sean A. Irvine, <a href="https://github.com/archmageirvine/joeis/blob/master/src/irvine/oeis/a390/A390670.java">Java program</a> (github)

%H John Tyler Rascoe, <a href="/A390670/a390670.py.txt">Python code</a>.

%H Christian Sievers, <a href="/A390670/a390670.lp.txt">clingo program</a>

%e Triangle begins:

%e    k=0     1    2    3    4   5  6

%e  n=0 1;

%e  n=1 0,    1;

%e  n=2 0,    1,   1;

%e  n=3 0,    2,   4,   1;

%e  n=4 0,    9,  15,   9,   1;

%e  n=5 0,   80,  79,  64,  16,  1;

%e  n=6 0, 1226, 705, 471, 185, 25, 1;

%e ...

%e The word (3,2,4,1,3,2) has subset C of largest size 1 and removing any of the repeated letters will change that, so this word is counted under T(4,1) = 9.

%e T(3,1) = 2: (2,3,1,2), (3,2,1).

%e T(3,2) = 4: (1,3,2), (2,1,3), (2,3,1), (3,1,2).

%e T(3,3) = 1: (1,2,3).

%o (Python) # see links

%o (clingo) % see links

%Y Cf. A000290 (empirical 2nd diagonal).

%Y Cf. A386891, A387336.

%K nonn,tabl,more

%O 0,8

%A _John Tyler Rascoe_, Nov 14 2025

%E a(28)-a(44) from _Sean A. Irvine_, Nov 28 2025

%E a(45)-a(54) (row n=9) from _Christian Sievers_, Nov 30 2025