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%I #21 Jan 27 2019 02:59:04
%S 1,2,3,1,4,1,5,1,2,6,1,2,1,7,1,2,3,1,8,1,2,3,1,1,9,1,2,3,4,1,1,2,10,1,
%T 2,3,4,1,1,1,2,1,11,1,2,3,4,5,1,1,1,2,2,1,12,1,2,3,4,5,1,1,1,1,2,2,3,
%U 1,1,13,1,2,3,4,5,6,1,1,1,1,2,2,2,3,1,1,1,14,1,2,3,4,5,6,1,1,1,1,1,2,2,2,3,3,1,1,1,1,2,15,1,2,3,4,5,6,7,1,1,1,1,1,2,2,2,2,3,3,4,1,1,1,1,1,2,1
%N Irregular triangular array T(n, k) giving in row n the base of the Ferrers diagram of the k-th partition of n into distinct parts. The partitions of n are taken in Abramowitz-Stegun order but with decreasing parts. T(n, k) is the smallest part of the k-th partition of n into distinct parts.
%C The length of row n of this irregular triangular array is A000009(n).
%C For the Abramowitz-Stegun order of partitions see an Apr 04 2011 comment on A036036.
%C The sum of the numbers of row n is A092265(n).
%C See the Hardy (H) and Hardy-Wright (H-W) references, where the base is called beta. The companion array is A277231 giving the slopes (called sigma) of these partitions with distinct parts. These beta and sigma numbers play a role in an elementary proof of Euler's pentagonal-number theorem (pp. 284-287 in (H-W), and pp. 83-85 in (H)) by F. Franklin from 1881. See also MacMahon and Charalambides.
%C The base of the Ferrers diagram of the k-th partition of n into distinct parts (in the mentioned order) is the number of nodes in the last row, the smallest part of the partition.
%C The slope of such a partition is the number of nodes on the NE-SW diagonal through the last node of the first row of the Ferrers diagram. (The name may be misleading. The usual slope of the NE-SW diagonal is of course 1).
%C The number of parts m of these partitions is from m = 1, 2, ..., A003056(n).
%D Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, pp. 389-391, 396, 595.
%D G. H. Hardy, Ramanujan, AMS Chelsea Publ., Providence, RI, 2002, pp. 83-85.
%D G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 5th ed., Clarendon Press, Oxford, 2003, pp. 284-287.
%D P. A. MacMahon, Combinatory Analysis, Vol. II, Chelsea Publishing Company, New York, 1960, pp. 21-23.
%H F. Franklin, <a href="https://gallica.bnf.fr/ark:/12148/bpt6k7351t/f447.image">Sur le développement du produit infini (1-x) (1-x^2) (1-x^3) ...</a>, Comptes Rendus de l'Académie des Sciences, Paris, 92 (1881) 448-450.
%F T(n, k) is the smallest part of the k-th partition of n into distinct parts. n >=1. k=1, 2, ..., A000009(n). Partitions appear in Abramowitz-Stegun order.
%e The irregular triangle begins (brackets separate partitions with equal number of parts m = 1, 2, 3,..., A003056(n)):
%e n\k 1 2 3 4 5 6 7 8 9 10 ...
%e 1: [1]
%e 2: [2]
%e 3: [3] [1]
%e 4: [4] [1]
%e 5: [5] [1, 2]
%e 6: [6] [1, 2] [1]
%e 7: [7] [1, 2, 3] [1]
%e 8: [8] [1, 2, 3] [1, 1]
%e 9: [9] [1, 2, 3, 4] [1, 1, 2]
%e 10: [10] [1, 2, 3, 4] [1, 1, 1, 2] [1]
%e ...
%e n = 11: [11] [1, 2, 3, 4, 5] [1, 1, 1, 2, 2] [1],
%e n = 12: [12] [1, 2, 3, 4, 5] [1, 1, 1, 1, 2, 2, 3] [1, 1],
%e n = 13: [13] [1, 2, 3, 4, 5, 6] [1, 1, 1, 1, 2, 2, 2, 3] [1, 1, 1],
%e n = 14: [14] [1, 2, 3, 4, 5, 6] [1, 1, 1, 1, 1, 2, 2, 2, 3, 3] [1, 1, 1, 1, 2],
%e n = 15: [15] [1, 2, 3, 4, 5, 6, 7] [1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4] [1, 1, 1, 1, 1, 2] [1].
%e ----------------------------------------
%e The partition of n = 5 + 4 + 1 = 10 has base 1 and slope 2 (beta < sigma):
%e o o o o o
%e o o o o
%e o
%e The partition of n = 5 + 3 + 1 = 9 has base 1 and slope 1 (beta = sigma):
%e o o o o o
%e o o o
%e o
%e The partition of n = 5 + 3 + 2 = 10 has base 2 and slope 1 (beta > sigma):
%e o o o o o
%e o o o
%e o o
%e ------------------------------------------
%e The partitions of n = 6 with m = 1, 2, and 3, (3 = A003056(6)) distinct parts are:
%e [6], [[5, 1], [4, 2]], [3, 2, 1], with base numbers in row n=6: [6] [1, 2] [1]
%e and slope numbers in row n=6 of A277231:
%e [1] [1, 1] [3].
%t Table[Function[w, Flatten@ Map[Function[k, Min /@ Select[w, Length@ # == k &]], Range@ Max@ Map[Length, w]]]@ Select[ DeleteCases[ IntegerPartitions@ n, w_ /; MemberQ[Differences@ w, 0]], Length@ # <= Floor[(Sqrt[1 + 8 n] - 1)/2] &], {n, 15}] // Flatten (* _Michael De Vlieger_, Oct 26 2016 *)
%Y Cf. A000009, A003056, A092265, A277231.
%K nonn,tabf,easy
%O 1,2
%A _Wolfdieter Lang_, Oct 21 2016
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