This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
Triangle begins: [1]; [2],[1]; [3],[1],[1]; [4],[2,2],[1],[1],[1]; [5],[3,2],[1],[1],[1],[1],[1]; [6],[3,3],[4,2],[2,2,2],[1],[1],[1],[1],[1],[1],[1]; [7],[4,3],[5,2],[3,2,2],[1],[1],[1],[1],[1],[1],[1],[1],[1],[1],[1]; ... The illustration of the three views of the section model of partitions (version "tree" with seven sections) shows the connection between several sequences. --------------------------------------------------------- Partitions A194805 Table 1.0 . of 7 p(n) A194551 A135010 --------------------------------------------------------- 7 15 7 7 . . . . . . 4+3 4 4 . . . 3 . . 5+2 5 5 . . . . 2 . 3+2+2 3 3 . . 2 . 2 . 6+1 11 6 1 6 . . . . . 1 3+3+1 3 1 3 . . 3 . . 1 4+2+1 4 1 4 . . . 2 . 1 2+2+2+1 2 1 2 . 2 . 2 . 1 5+1+1 7 1 5 5 . . . . 1 1 3+2+1+1 1 3 3 . . 2 . 1 1 4+1+1+1 5 4 1 4 . . . 1 1 1 2+2+1+1+1 2 1 2 . 2 . 1 1 1 3+1+1+1+1 3 1 3 3 . . 1 1 1 1 2+1+1+1+1+1 2 2 1 2 . 1 1 1 1 1 1+1+1+1+1+1+1 1 1 1 1 1 1 1 1 1 . 1 --------------- . *<------- A000041 -------> 1 1 2 3 5 7 11 . A182712 -------> 1 0 2 1 4 3 . A182713 -------> 1 0 1 2 2 . A182714 -------> 1 0 1 1 . 1 0 1 . A141285 A182703 1 0 . A182730 A182731 1 --------------------------------------------------------- . A138137 --> 1 2 3 6 9 15.. --------------------------------------------------------- . A182746 <--- 4 . 2 1 0 1 2 . 4 ---> A182747 --------------------------------------------------------- . . A182732 <--- 6 3 4 2 1 3 5 4 7 ---> A182733 . . . . . 1 . . . . . . . . 2 1 . . . . . . 3 . . 1 2 . . . . Table 2.0 . . 2 2 1 . . 3 . Table 2.1 . . . . . 1 2 2 . . . 1 . . . . . . A182982 A182742 A194803 A182983 A182743 . A182992 A182994 A194804 A182993 A182995 --------------------------------------------------------- . From _Omar E. Pol_, Sep 03 2013: (Start) Illustration of initial terms (n = 1..6). The table shows the six sections of the set of partitions of 6. Note that before the dissection the set of partitions was in the ordering mentioned in A026792. More generally, the six sections of the set of partitions of 6 also can be interpreted as the first six sections of the set of partitions of any integer >= 6. Illustration of initial terms: --------------------------------------- n j Diagram Parts --------------------------------------- . _ 1 1 |_| 1; . _ _ 2 1 |_ | 2, 2 2 |_| . 1; . _ _ _ 3 1 |_ _ | 3, 3 2 | | . 1, 3 3 |_| . . 1; . _ _ _ _ 4 1 |_ _ | 4, 4 2 |_ _|_ | 2, 2, 4 3 | | . 1, 4 4 | | . . 1, 4 5 |_| . . . 1; . _ _ _ _ _ 5 1 |_ _ _ | 5, 5 2 |_ _ _|_ | 3, 2, 5 3 | | . 1, 5 4 | | . . 1, 5 5 | | . . 1, 5 6 | | . . . 1, 5 7 |_| . . . . 1; . _ _ _ _ _ _ 6 1 |_ _ _ | 6, 6 2 |_ _ _|_ | 3, 3, 6 3 |_ _ | | 4, 2, 6 4 |_ _|_ _|_ | 2, 2, 2, 6 5 | | . 1, 6 6 | | . . 1, 6 7 | | . . 1, 6 8 | | . . . 1, 6 9 | | . . . 1, 6 10 | | . . . . 1, 6 11 |_| . . . . . 1; ... (End)
less[run1_, run2_] := (lg1 = run1 // Length; lg2 = run2 // Length; lg = Max[lg1, lg2]; r1 = If[lg1 == lg, run1, PadRight[run1, lg, 0]]; r2 = If[lg2 == lg, run2, PadRight[run2, lg, 0]]; Order[r1, r2] != -1); row[n_] := Join[Array[1 &, {PartitionsP[n - 1]}], Sort[Reverse /@ Select[IntegerPartitions[n], FreeQ[#, 1] &], less]] // Flatten // Reverse; Table[row[n], {n, 1, 9}] // Flatten (* Jean-François Alcover, Jan 15 2013 *)
Table[Reverse/@Reverse@DeleteCases[Sort@PadRight[Reverse/@Cases[IntegerPartitions[n], x_ /; Last[x]!=1]], x_ /; x==0, 2]~Join~ConstantArray[{1}, PartitionsP[n - 1]], {n, 1, 9}] // Flatten (* Robert Price, May 11 2020 *)
Written as a triangle T(j,k) the sequence begins: 1; 2; 3; 2, 4; 3, 5; 2, 4, 3, 6; 3, 5, 4, 7; 2, 4, 3, 6, 5, 4, 8; 3, 5, 4, 7, 3, 6, 5, 9; 2, 4, 3, 6, 5, 4, 8, 4, 7, 6, 5, 10; 3, 5, 4, 7, 3, 6, 5, 9, 5, 4, 8, 7, 6, 11; ... ------------------------------------------ n A000041 a(n) ------------------------------------------ 1 = p(1) 1 2 = p(2) 2 . 3 = p(3) . 3 4 2 . 5 = p(4) 4 . 6 . 3 7 = p(5) . 5 8 2 . 9 4 . 10 3 . 11 = p(6) 6 . 12 . 3 13 . 5 14 . 4 15 = p(7) . 7 ... From _Omar E. Pol_, Aug 22 2013: (Start) Illustration of initial terms (n = 1..11) in three ways: as the largest parts of the partitions of 6 (see A026792), also as the largest parts of the regions of the diagram, also as the diagonal of triangle. By definition of "region" the largest part of the n-th region is also the largest part of the n-th partition (see below): -------------------------------------------------------- . Diagram Triangle in which Partitions of regions rows are partitions of 6 and partitions and columns are regions -------------------------------------------------------- . _ _ _ _ _ _ 6 _ _ _ | 6 3+3 _ _ _|_ | 3 3 4+2 _ _ | | 4 2 2+2+2 _ _|_ _|_ | 2 2 2 5+1 _ _ _ | | 5 1 3+2+1 _ _ _|_ | | 3 1 1 4+1+1 _ _ | | | 4 1 1 2+2+1+1 _ _|_ | | | 2 2 1 1 3+1+1+1 _ _ | | | | 3 1 1 1 2+1+1+1+1 _ | | | | | 2 1 1 1 1 1+1+1+1+1+1 | | | | | | 1 1 1 1 1 1 ... The equivalent sequence for compositions is A001511. Explanation: for the positive integer j the diagram of regions of the set of compositions of j has 2^(j-1) regions. The largest part of the n-th region is A001511(n). The number of parts is A006519(n). On the other hand the diagram of regions of the set of partitions of j has A000041(j) regions. The largest part of the n-th region is a(n) = A001511(A228354(n)). The number of parts is A194446(n). Both diagrams have j sections. The diagram for partitions can be interpreted as one of the three views of a three dimensional diagram of compositions in which the rows of partitions are in orthogonal direction to the rest. For the first five sections of the diagrams see below: -------------------------------------------------------- . Diagram Diagram . of regions of regions . and compositions and partitions --------------------------------------------------------- . j = 1 2 3 4 5 j = 1 2 3 4 5 --------------------------------------------------------- n A001511 A228354 a(n) --------------------------------------------------------- 1 1 _| | | | | ............ 1 1 _| | | | | 2 2 _ _| | | | ............ 2 2 _ _| | | | 3 1 _| | | | ......... 4 3 _ _ _| | | 4 3 _ _ _| | | ../ ....... 6 2 _ _| | | 5 1 _| | | | / ....... 8 4 _ _ _ _| | 6 2 _ _| | | ../ / .... 12 3 _ _ _| | 7 1 _| | | / / . 16 5 _ _ _ _ _| 8 4 _ _ _ _| | ../ / / 9 1 _| | | | / / 10 2 _ _| | | / / 11 1 _| | | / / 12 3 _ _ _| | ../ / 13 1 _| | | / 14 2 _ _| | / 15 1 _| | / 16 5 _ _ _ _ _| ../ ... Also we can draw an infinite Dyck path in which the n-th odd-indexed line segment has a(n) up-steps and the n-th even-indexed line segment has A194446(n) down-steps. Note that the height of the n-th largest peak between two successive valleys at height 0 is also the partition number A000041(n). See below: . 5 . /\ 3 . 4 / \ 4 /\ . /\ / \ /\ / . 3 / \ 3 / \ / \/ . 2 /\ 2 / \ /\/ \ 2 / . 1 /\ / \ /\/ \ / \ /\/ . /\/ \/ \/ \/ \/ . .(End)
Last/@DeleteCases[DeleteCases[Sort@PadRight[Reverse/@IntegerPartitions[13]], x_ /; x == 0, 2], {}] (* updated _Robert Price, May 15 2020 *)
Array begins: 3,2,2,2,2,2,2,2,2,2,2, 5,2,2,2,2,2,2,2,2,2, 4,3,2,2,2,2,2,2,2,2, 7,2,2,2,2,2,2,2,2, 3,3,3,2,2,2,2,2,2,2, 6,3,2,2,2,2,2,2,2, 5,4,2,2,2,2,2,2,2, 9,2,2,2,2,2,2,2, 5,3,3,2,2,2,2,2,2, 4,4,3,2,2,2,2,2,2, 8,3,2,2,2,2,2,2, 7,4,2,2,2,2,2,2, 6,5,2,2,2,2,2,2, 11,2,2,2,2,2,2, 4,3,3,3,2,2,2,2,2, 7,3,3,2,2,2,2,2, 6,4,3,2,2,2,2,2, 5,5,3,2,2,2,2,2, 10,3,2,2,2,2,2, 5,4,4,2,2,2,2,2, 9,4,2,2,2,2,2, 8,5,2,2,2,2,2, 7,6,2,2,2,2,2,
cmpL := proc(a,b) local i ; for i from 1 to min(nops(a),nops(b)) do if op(i,a) < op(i,b) then return -1 ; elif op(i,a) > op(i,b) then return 1 ; end if; end do; if nops(a) > nops(b) then return 1; elif nops(a) < nops(b) then return -1; else return 0; end if; end proc: pShellMin := proc(p) local idx,j; idx := 1 ; for j from 2 to nops(p) do if cmpL( op(j,p),op(idx,p)) < 0 then idx := j; end if; end do; return idx ; end proc: A141285rowf := proc(n) local p; if n <= 1 then [n] ; else psort := [] ; p := combinat[partition](n) ; while nops(p) > 0 do m := pShellMin(p) ; mmi := min(op(op(m,p))) ; if mmi > 1 then mma := max(op(op(m,p))) ; psort := [op(psort),sort(op(m,p),`>`)] ; end if; p := subsop(m=NULL,p) ; end do: psort ; end if; end proc: for n from 1 to 17 by 2 do shl := A141285rowf(n) ; for r in shl do for k in r do printf("%d,",k) ; end do: printf("\n") ; end do: printf("\n") ; end do: # R. J. Mathar, Dec 03 2010
Triangle begins: 0, 2, 2, 4, 2, 4, 3, 6, 2, 4, 3, 6, 5, 4, 8, 2, 4, 3, 6, 5, 4, 8, 4, 7, 6, 5, 10, 2, 4, 3, 6, 5, 4, 8, 4, 7, 6, 5, 10, 3, 6, 5, 9, 4, 8, 7, 6, 12
Written as a triangle begins: 1; 2; 3; 4; 5; 3,6; 4,7; 5,4,8; 3,6,5,9; 4,7,6,5,10; 5,4,8,7,6,11; 3,6,5,9,4,8,7,6,12; 4,7,6,5,10,5,9,8,7,13; 5,4,8,7,6,11,6,5,10,9,8,7,14; ... Row n has length A008483(n), if n >= 3.
Join[{1},Table[Drop[l = Last/@DeleteCases[Sort@PadRight[Reverse /@ Cases[IntegerPartitions[n], x_ /; Last[x] != 1]], x_ /; x == 0, 2], First@FirstPosition[l, n - 2, {0}]], {n, 2, 15}]] // Flatten (* Robert Price, May 15 2020 *)
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