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Triangle read by rows: T(j,k) is the total number of appearances of the smallest parts in the j-th partition of n, with partitions as nonincreasing lists of parts in lexicographic order.
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%I #17 Dec 13 2015 01:11:22

%S 1,2,1,3,1,1,4,2,1,2,1,5,3,2,1,1,1,1,6,4,3,2,2,1,1,3,1,2,1,7,5,4,3,3,

%T 2,2,1,1,1,1,2,1,1,1,8,6,5,4,4,3,3,2,2,2,2,1,1,1,1,4,2,1,1,1,2,1,9,7,

%U 6,5,5,4,4,3,3,3,3,2,2,2,2,1,1,1,1,1,1,1,3,2,1,1,3,1,1,1

%N Triangle read by rows: T(j,k) is the total number of appearances of the smallest parts in the j-th partition of n, with partitions as nonincreasing lists of parts in lexicographic order.

%C The sum of row n equals spt(n) = A092269(n), the smallest part partition function.

%e For n = 5:

%e ------------------------------------------

%e . number of

%e Partitions of 5 smallest parts

%e ------------------------------------------

%e 1 + 1 + 1 + 1 + 1 5

%e 2 + 1 + 1 + 1 3

%e 3 + 1 + 1 2

%e 2 + 2 + 1 1

%e 4 + 1 1

%e 3 + 2 1

%e 5 1

%e ------------------------------------------

%e So row 5 is [5, 3, 2, 1, 1, 1, 1]. The sum of row 5 is 5+3+2+1+1+1+1 = spt(5) = A092269(n) = 14.

%e .

%e Written as an irregular triangle begins:

%e 1;

%e 2,1;

%e 3,1,1;

%e 4,2,1,2,1;

%e 5,3,2,1,1,1,1;

%e 6,4,3,2,2,1,1,3,1,2,1;

%e 7,5,4,3,3,2,2,1,1,1,1,2,1,1,1;

%e 8,6,5,4,4,3,3,2,2,2,2,1,1,1,1,4,2,1,1,1,2,1;

%e 9,7,6,5,5,4,4,3,3,3,3,2,2,2,2,1,1,1,1,1,1,1,3,2,1,1,3,1,1,1;

%Y Column 1 is A000027. Row n has length A000041(n). Row sums give A092269.

%Y Cf. A209514, A220504.

%K nonn,tabf

%O 1,2

%A _Omar E. Pol_, Jan 20 2013