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Arises in partition theory.
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%I #8 Jul 17 2023 06:04:21

%S 1,2,3,4,4,5,6,5,6,6,7,8,6,7,7,8,8,9,10

%N Arises in partition theory.

%C The number of entries <= n gives A000041(n) (the partition numbers). The length of column n is also A000041(n).

%e Write 0..n as column indices. Under each column write a number for each word of length n+1 of nonisomorphic ballot sequences on 2..(n+1), where the number is n+the number of distinct elements of 2..(n+1). So;

%e 0 1 2 3 4 5

%e 1 2 3 4 5 6

%e ... 4 5 6 7

%e ..... 6 6 7

%e ....... 7 8

%e ....... 8 8

%e ......... 9

%e ......... 10

%e e.g. for n=5, consider 22222, 22223, 22233, 22234, 22334, 22345, 23456, giving 6,7,7,8,8,9,10.

%e The sequence reads the columns in turn.

%K hard,nonn,tabf

%O 0,2

%A _Jon Perry_, Mar 26 2004