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%I #13 May 22 2020 13:22:49
%S 1,1,2,1,1,3,1,1,1,2,2,4,1,1,1,1,1,2,3,5,1,1,1,1,1,1,1,2,2,2,3,3,2,4,
%T 6,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,2,5,7,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
%U 1,2,2,2,2,2,3,3,2,2,4,4,4,3,5,2,6,8,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
%N A shell model of partitions. Triangle read by rows: row n lists the parts of the last section of the set of partitions of n.
%C The Integrated Diagram of Partitions is a shell model of partitions of a number. Partitions of n contains all partitions of the previous numbers. The number of shells of the partitions of n is equal to n. The number of parts of the last section of the set of partitions of n is A138137(n)=A006128(n)-A006128(n-1) and equal to the number of terms of row n. The number of terms of row n that are equal to 1 is A000041(n-1). The last term of row n is n. The shell model of partitions has several 2D and 3D versions.
%H Robert Price, <a href="/A138138/b138138.txt">Table of n, a(n) for n = 1..4630, 20 rows.</a>
%e ........................................
%e .. Integrated Diagram of Partitions ...
%e ........... for n = 1 to 9 ............
%e .......................................
%e Partition number \ n = 1 2 3 4 5 6 7 8 9
%e ........................................
%e .1) A000041(1)= 1 .... 1 1 1 1 1 1 1 1 1
%e .2) A000041(2)= 2 .... . 2 1 1 1 1 1 1 1
%e .3) A000041(3)= 3 .... . . 3 1 1 1 1 1 1
%e .4) .................. . 2 . 2 1 1 1 1 1
%e .5) A000041(4)= 5 .... . . . 4 1 1 1 1 1
%e .6) .................. . . 3 . 2 1 1 1 1
%e .7) A000041(5)= 7 .... . . . . 5 1 1 1 1
%e .8) .................. . 2 . 2 . 2 1 1 1
%e .9) .................. . . 3 . . 3 1 1 1
%e 10) .................. . . . 4 . 2 1 1 1
%e 11) A000041(6)=11 .... . . . . . 6 1 1 1
%e 12) .................. . . 3 . 2 . 2 1 1
%e 13) .................. . . . 4 . . 3 1 1
%e 14) .................. . . . . 5 . 2 1 1
%e 15) A000041(7)=15 .... . . . . . . 7 1 1
%e 16) .................. . 2 . 2 . 2 . 2 1
%e 17) .................. . . 3 . . 3 . 2 1
%e 18) .................. . . . 4 . 2 . 2 1
%e 19) .................. . . . 4 . . . 4 1
%e 20) .................. . . . . 5 . . 3 1
%e 21) .................. . . . . . 6 . 2 1
%e 22) A000041(8)=22 .... . . . . . . . 8 1
%e 23) .................. . . 3 . 2 . 2 . 2
%e 24) .................. . . 3 . . 3 . . 3
%e 25) .................. . . . 4 . . 3 . 2
%e 26) .................. . . . . 5 . 2 . 2
%e 27) .................. . . . . 5 . . . 4
%e 28) .................. . . . . . 6 . . 3
%e 29) .................. . . . . . . 7 . 2
%e 30) A000041(9)=30 .... . . . . . . . . 9
%e .......................................
%e Triangle begins:
%e 1
%e 1,2
%e 1,1,3,
%e 1,1,1,2,2,4
%e 1,1,1,1,1,2,3,5
%e 1,1,1,1,1,1,1,2,2,2,3,3,2,4,6
%e 1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,4,2,5,7
%e 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,3,2,2,4,4,4,3,5,2,6,8
%e 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3,3,3,3,2,3,4,2,2,5,4,5,3,6,2,7,9
%t Table[ConstantArray[{1}, PartitionsP[n - 1]] ~Join~ Reverse@Flatten@Cases[IntegerPartitions[n], x_ /; Last[x] != 1], {n, 8}] // Flatten (* _Robert Price_, May 22 2020 *)
%Y Cf. A000041, A006128, A138137. See A135010 for another version.
%K nonn,tabf,less
%O 1,3
%A _Omar E. Pol_, Mar 16 2008, Mar 25 2008
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