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%I #78 Oct 02 2013 15:58:41
%S 1,2,1,1,3,2,1,1,1,1,5,4,1,2,2,1,1,2,1,1,7,6,1,4,2,1,2,3,1,1,1,2,2,1,1
%N Tetrahedron in which the n-th slice is also one of the three views of the shell model of partitions of A207380 with n shells.
%C Each slice of the tetrahedron is a triangle, thus the number of elements in the n-th slice is A000217(n). The slices are perpendicular to the slices of A026792. Each element of the n-th slice equals the volume of a column of the shell model of partitions with n shells. The sum of each row of the n-th slice is A000041(n). The sum of all elements of the n-th slice is A066186(n).
%C It appears that the triangle formed by the last row of each slice gives A008284 and A058398.
%C It appears that the triangle formed by the first column of each slice gives A058399.
%C Also consider a vertical rectangle on the infinite square grid with shorter side = n and longer side = p(n) = A000041(n). Each row of rectangle represents a partition of n. Each part of each partition of n is a horizontal rectangle with shorter side = 1 and longer side = k, where k is the size of the part. It appears that T(n,k,j) is also the number of k-th parts of all partitions of n in the j-th column of rectangle.
%e --------------------------------------------------------
%e Illustration of first five
%e slices of the tetrahedron Row sum
%e --------------------------------------------------------
%e . 1, 1
%e . 2, 2
%e . 1, 1, 2
%e . 3, 3
%e . 2, 1, 3
%e . 1, 1, 1, 3
%e . 5, 5
%e . 4, 1, 5
%e . 2, 2, 1, 5
%e . 1, 2, 1, 1, 5
%e . 7, 7
%e . 6, 1, 7
%e . 4, 2, 1, 7
%e . 2, 3, 1, 1, 7
%e . 1, 2, 2, 1, 1, 7
%e --------------------------------------------------------
%e . 1, 3, 1, 6, 2, 1,12, 5, 2, 1,20, 8, 4, 2, 1,
%e .
%e Written as a triangle begins:
%e 1;
%e 2, 1, 1;
%e 3, 2, 1, 1, 1, 1;
%e 5, 4, 1, 2, 2, 1, 1, 2, 1, 1;
%e 7, 6, 1, 4, 2, 1, 2, 3, 1, 1, 1, 2, 2, 1, 1;
%e In which row sums give A066186.
%Y Column sums give A181187. Main diagonal gives A210765. Another version is A209918.
%Y Cf. A000041, A000217, A002260, A004736, A008284, A026792, A058398, A058399, A066186, A135010, A182703, A182715, A207380.
%K nonn,tabf,more
%O 1,2
%A _Omar E. Pol_, Mar 25 2012
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