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A211004 Number of distinct regions in the set of partitions of n. 0

%I #40 Mar 12 2015 19:38:22

%S 1,2,3,5,7,9,12,15,18,22,26,30,35,40,45,51

%N Number of distinct regions in the set of partitions of n.

%C The number of regions in the set of partitions of n equals the number of partitions of n. The sequence counts only the distinct regions. For the definition of "regions of the set of partitions of n" (or more simply "regions of n") see A206437.

%C Is this the same as A001840 for all positive integers? If not, where is the first place these sequences differ?

%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpa406.jpg">Illustration of initial terms</a>

%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpar02.jpg">Illustration of the seven regions of 5</a>

%e For n = 6 the 11 regions of 6 are [1], [2,1], [3,1,1], [2], [4,2,1,1,1], [3], [5,2,1,1,1,1,1], [2], [4,2], [3], [6,3,2,2,1,1,1,1,1,1,1]. These number are the first A006128(6) terms of triangle A206437 in which the first A000041(6) rows are the 11 regions of 6. We can see that the 8th region is equal to the 4th region: [2] = [2]. Also the 10th region is equal to the 6th region: [3] = [3]. There are two repeated regions, therefore a(6) = A000041(6) - 2 = 11 - 2 = 9.

%Y Cf. A000041, A006128, A135010, A141285, A182244, A186114, A186412, A187219, A193870, A194446, A194447, A206437, A211009.

%K nonn,more

%O 1,2

%A _Omar E. Pol_, Oct 22 2012

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