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: 2, 2, 4, 2, 2, 3, 3, 6, 2, 2, 2, 2, 3, 5, 4, 4, 8, 2, 2, 2, 2, 2, 2, 2, 3, 3, 4, 3, 7, 4, 6, 5, 5, 10
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 *)
Illustration of three arrangements of the last section of the set of partitions of 7, or more generally the 7th section of the set of partitions of any integer >= 7: . _ _ _ _ _ _ _ . (7) (7) |_ _ _ _ | . (4+3) (4+3) |_ _ _ _|_ | . (5+2) (5+2) |_ _ _ | | . (3+2+2) (3+2+2) |_ _ _|_ _|_ | . (1) (1) | | . (1) (1) | | . (1) (1) | | . (1) (1) | | . (1) (1) | | . (1) (1) | | . (1) (1) | | . (1) (1) | | . (1) (1) | | . (1) (1) | | . (1) (1) |_| . ---------------- . 19,8,5,3,2,1,1 --> Row 7 of triangle A207031. . |/|/|/|/|/|/| . 11,3,2,1,1,0,1 --> Row 7 of this triangle. . Note that the "head" of the last section is formed by the partitions of 7 that do not contain 1 as a part. The "tail" is formed by A000041(7-1) parts of size 1. The number of rows (or zones) is A000041(7) = 15. The last section of the set of partitions of 7 contains eleven 1's, three 2's, two 3's, one 4, one 5, there are no 6's and it contains one 7. So, for k = 1..7, row 7 gives: 11, 3, 2, 1, 1, 0, 1. Triangle begins: 1; 1, 1; 2, 0, 1; 3, 2, 0, 1; 5, 1, 1, 0, 1; 7, 4, 2, 1, 0, 1; 11, 3, 2, 1, 1, 0, 1; 15, 8, 3, 3, 1, 1, 0, 1; 22, 7, 6, 2, 2, 1, 1, 0, 1; 30, 15, 6, 5, 3, 2, 1, 1, 0, 1; 42, 15, 10, 5, 4, 2, 2, 1, 1, 0, 1; 56, 27, 14, 10, 5, 5, 2, 2, 1, 1, 0, 1; ...
p:= (f, g)-> zip((x, y)-> x+y, f, g, 0):
b:= proc(n,i) option remember; local g;
if n=0 then [1]
elif n<2 or i<2 then [0]
else g:= `if`(i>n, [0], b(n-i, i));
p(p([0$j=2..i, g[1]], b(n, i-1)), g)
fi
end:
h:= proc(n) option remember;
`if`(n=0, 1, b(n, n)[1]+h(n-1))
end:
T:= proc(n) h(n-1), b(n, n)[2..n][] end:
seq(T(n), n=1..20); # Alois P. Heinz, Feb 19 2012
p[f_, g_] := Plus @@ PadRight[{f, g}]; b[n_, i_] := b[n, i] = Module[{g}, Which[n == 0, {1}, n<2 || i<2, {0}, True, g = If [i>n, {0}, b[n-i, i]]; p[p[Append[Array[0&, i-1], g[[1]]], b[n, i-1]], g]]]; h[n_] := h[n] = If[n == 0, 1, b[n, n][[1]] + h[n-1]]; t[n_] := {h[n-1], Sequence @@ b[n, n][[2 ;; n]]}; Table[t[n], {n, 1, 20}] // Flatten (* Jean-François Alcover, Jan 16 2014, after Alois P. Heinz's Maple code *)
Table[{PartitionsP[n-1]}~Join~Table[Count[Flatten@Cases[IntegerPartitions[n], x_ /; Last[x] != 1], k], {k,2,n}], {n,1,12}] // Flatten (* 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
For n=7 the partitions of 7 that do not contain "1" as a part are 7 4 + 3 5 + 2 3 + 2 + 2 Then remove one copy of the smallest part of every partition. The rest are the emergent parts: ., 4, . 5, . 3, 2, . The sum of these parts is 4 + 5 + 3 + 2 = 14, so a(7)=14. For n=10 the illustration in the link shows the location of the emergent parts (colored yellow and green) and the location of the filler parts (colored blue) in the last section of the set of partitions of 10.
b:= proc(n, i) option remember;
if n<0 then 0
elif n=0 then 1
elif i<2 then 0
else b(n, i-1) +b(n-i, i)
fi
end:
c:= proc(n, i, k) option remember;
if n<0 then 0
elif n=0 then k
elif i<2 then 0
else c(n, i-1, k) +c(n-i, i, i)
fi
end:
a:= n-> n*b(n, n) - c(n, n, 0):
seq(a(n), n=1..40); # Alois P. Heinz, Dec 01 2010
f[n_]:=Total[Flatten[Most/@Select[IntegerPartitions[n],!MemberQ[#,1]&]]]; Table[f[i],{i,50}] (* Harvey P. Dale, Dec 28 2010 *)
b[n_, i_] := b[n, i] = Which[n<0, 0, n==0, 1, i<2, 0, True, b[n, i-1] + b[n - i, i]]; c[n_, i_, k_] := c[n, i, k] = Which[n<0, 0, n==0, k, i<2, 0, True, c[n, i-1, k] + c[n-i, i, i]]; a[n_] := n*b[n, n] - c[n, n, 0]; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Oct 08 2015, after Alois P. Heinz *)
b:= proc(n, i) option remember;
if n<0 then 0
elif n=0 then 1
elif i<2 then 0
else b(n, i-1) +b(n-i, i)
fi
end:
a:= n-> b(2*n, 2*n):
seq(a(n), n=0..40); # Alois P. Heinz, Dec 01 2010
Table[Count[IntegerPartitions[2 n -1], p_ /; MemberQ[p, Length[p]]], {n, 20}] (* Clark Kimberling, Mar 02 2014 *)
b[n_, i_] := b[n, i] = Which[n<0, 0, n==0, 1, i<2, 0, True, b[n, i-1] + b[n-i, i]]; a[n_] := b[2*n, 2*n]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Sep 21 2015, after Alois P. Heinz *)
a[n_] := PartitionsP[2*n] - PartitionsP[2*n - 1]; Table[a[n], {n, 0, 40}] (* George Beck, Jun 05 2017 *)
a(n)=numbpart(2*n)-numbpart(2*n-1) \\ Charles R Greathouse IV, Jun 06 2017
b:= proc(n,i) option remember;
if n<0 then 0
elif n=0 then 1
elif i<2 then 0
else b(n, i-1) +b(n-i, i)
fi
end:
a:= n-> b(2*n+1, 2*n+1):
seq(a(n), n=0..40); # Alois P. Heinz, Dec 01 2010
f[n_] := Table[PartitionsP[2 k + 1] - PartitionsP[2 k], {k, 0, n}] (* George Beck, Aug 14 2011 *)
(* also *)
Table[Count[IntegerPartitions[2 n], p_ /; MemberQ[p, Length[p]]], {n, 20}] (* Clark Kimberling, Mar 02 2014 *)
b[n_, i_] := b[n, i] = Which[n<0, 0, n == 0, 1, i<2, 0, True, b[n, i-1] + b[n-i, i]]; a[n_] := b[2*n+1, 2*n+1]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Aug 29 2016, after Alois P. Heinz *)
a(5)=64 because the 5th subshell of the head of the last section of any even integer >= 10 looks like this: (10 . . . . . . . . . ) ( 5 . . . . 5 . . . . ) ( 6 . . . . . 4 . . . ) ( 7 . . . . . . 3 . . ) ( 4 . . . 3 . . 3 . . ) . (2 . ) . (2 . ) . (2 . ) . (2 . ) . (2 . ) . (2 . ) . (2 . ) There are 17 parts whose sum is 10+5+5+6+4+7+3+4+3+3+2+2+2+2+2+2+2 = 10*5 + 2*7 = 64, so a(5)=64.
For the numbers 1..6 the shell model of partitions has 6 shells. The model as a table looks like this: 1 1 1 1 1 1 2 . 1 1 1 1 3 . . 1 1 1 2 . 2 . 1 1 4 . . . 1 1 3 . . 2 . 1 5 . . . . 1 2 . 2 . 2 . 4 . . . 2 . 3 . . 3 . . 6 . . . . . Then replace the dots by zeros. Remarks: one number by column, for example 23 is located only in a column, not in two columns. The table looks like this: 1 1 1 1 1 1 2 0 1 1 1 1 3 0 0 1 1 1 2 0 2 0 1 1 4 0 0 0 1 1 3 0 0 2 0 1 5 0 0 0 0 1 2 0 2 0 2 0 4 0 0 0 2 0 3 0 0 3 0 0 6 0 0 0 0 0 Array begins: 1,1,1,1,1,1, 2,0,1,1,1, 3,0,0,1, 2,0,2, 4,0, 3,
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