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A182703 Triangle read by rows: T(n,k) = number of occurrences of k in the last section of the set of partitions of n. 89
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

For the definition of "section" of the set of partitions of n see A135010.

Also, column 1 gives the number of partitions of n-1. For k >= 2, row n lists the number of k's in all partitions of n that do not contain 1 as a part.

From Omar E. Pol, Feb 12 2012: (Start)

It appears that reversed rows converge to A002865.

It appears that row n is also the base of an isosceles triangle in which the column sums give the partition numbers A000041 in descending order starting with p(n-1) = A000041(n-1). Example for n = 7:

.

.         1,

.      1, 0, 1,

.   4, 2, 1, 0, 1,

11, 3, 2, 1, 1, 0, 1,

---------------------

11, 7, 5, 3, 2, 1, 1,

.

It appears that in row n starts an infinite trapezoid in which columns sums give always the number of partitions of n-1. Example for n = 7:

.

11, 3, 2, 1, 1, 0, 1,

.   8, 3, 3, 1, 1, 0, 1,

.      6, 2, 2, 1, 1, 0, 1,

.         5, 3, 2, 1, 1, 0, 1,

.            4, 2, 2, 1, 1, 0, 1,

.               5, 2, 2, 1, 1, 0,...

.                  4, 2, 2, 1, 1,...

.                     4, 2, 2, 1,...

.                        4, 2, 2,...

.                           4, 2,...

.                              4,...

.

The sum of any column is always p(7-1) = p(6) = A000041(6) = 11.

It appears that the first term of row n is one of the vertices of an infinite isosceles triangle in which column sums give the partition numbers A000041 in ascending order starting with p(n-1) = A000041(n-1). Example for n = 7:

11,

.    8,

.    7,  6,

.        6,  5,

.       10,  5,...

.           10,...

.           10,...

------------------

11, 15, 22, 30,...

(End)

It appears that row n lists the first differences of the row n of triangle A207031 together with 1 (as the final term of row n). - Omar E. Pol, Feb 26 2012

More generally T(n,k) is the number of occurrences of k in the n-th section of the set of partitions of any integer >= n. - Omar E. Pol, Oct 21 2013

LINKS

Alois P. Heinz, Rows n = 1..141, flattened

Omar E. Pol, Illustration of the section model of partitions, Figure 1 (n = 1..6), Figure 2 (2D view, n = 1..10), Figure 3 (3D view, n = 1..9)

FORMULA

It appears that T(n,k) = A207032(n,k) - A207032(n,k+2). - Omar E. Pol, Feb 26 2012

EXAMPLE

Illustration of three arrangements of the last section of the set of partitions of 7, or more generally the n-th 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;

...

MAPLE

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

MATHEMATICA

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-Fran├žois Alcover, Jan 16 2014, after Alois P. Heinz's Maple code *)

CROSSREFS

Row sums give A138137. Where records occur is A134869.

Columns 1-10: A000041, A182712-A182714, A206555-A206560.

Sub-triangles (1-11): A023531, A129186, A194702-A194710

Cf. A066633, A135010, A182742, A182743, A194812, A206563, A207031, A207032, A206437, A211025.

Sequence in context: A197707 A216220 A216235 * A231354 A197119 A124377

Adjacent sequences:  A182700 A182701 A182702 * A182704 A182705 A182706

KEYWORD

nonn,tabl,look

AUTHOR

Omar E. Pol, Nov 28 2010

STATUS

approved

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Last modified April 25 00:50 EDT 2014. Contains 240991 sequences.