login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A330369 Triangle read by rows: T(n,k) (1 <= k <= n) is the total number of right angles of size k in all partitions of n. 5
1, 0, 2, 0, 0, 3, 1, 0, 1, 4, 2, 0, 0, 2, 5, 3, 2, 0, 2, 3, 6, 4, 4, 0, 0, 4, 4, 7, 5, 6, 3, 0, 3, 6, 5, 8, 7, 8, 7, 0, 1, 6, 8, 6, 9, 9, 10, 11, 4, 0, 6, 9, 10, 7, 10, 13, 12, 15, 10, 0, 2, 11, 12, 12, 8, 11 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

This triangle has the property that it contains the triangle A049597, since if we replace with zeros the positive terms before the first zero in the row n of this triangle, we get the triangle A049597.

Hence the sum of the terms after the last zero in row n equals A000041(n), the number of partitions of n (see the Example section).

Observation: at least the first 11 terms of column 1 coincide with A188674 (using the same indices).

REFERENCES

G. E. Andrews, Theory of Partitions, Cambridge University Press, 1984, page 143 [Defines the right angles in the Ferrers graph of a partition. - N. J. A. Sloane, Nov 20 2020]

LINKS

Table of n, a(n) for n=1..66.

EXAMPLE

Triangle begins:

   1;

   0,  2;

   0,  0,  3;

   1,  0,  1,  4;

   2,  0,  0,  2,  5;

   3,  2,  0,  2,  3,  6;

   4,  4,  0,  0,  4,  4,  7;

   5,  6,  3,  0,  3,  6,  5,  8;

   7,  8,  7,  0,  1,  6,  8,  6,  9;

   9, 10, 11,  4,  0,  6,  9, 10,  7, 10;

  13, 12, 15, 10,  0,  2, 11, 12, 12,  8, 11;

Figure 1 below shows the Ferrers diagram of the partition of 24: [7, 6, 3, 3, 2, 1, 1, 1]. Figure 2 shows the right-angles diagram of the same partition. Note that in this last diagram we can see the size of the three right angles as follows: the first right angle has size 14 because it contains 14 square cells, the second right angle has size 8 and the third right angle has size 2.

.

.                                     Right-angles   Right

Part   Ferrers diagram         Part   diagram        angle

                                      _ _ _ _ _ _ _

  7    * * * * * * *             7   |  _ _ _ _ _ _|  14

  6    * * * * * *               6   | |  _ _ _ _|     8

  3    * * *                     3   | | | |           2

  3    * * *                     3   | | |_|

  2    * *                       2   | |_|

  1    *                         1   | |

  1    *                         1   | |

  1    *                         1   |_|

.

       Figure 1.                      Figure 2.

.

For n = 8 the partitions of 8 and their respective right-angles diagrams are as follows:

.

    _       _ _       _ _ _       _ _ _ _       _ _ _ _ _

  1| |8   2|  _|8   3|  _ _|8   4|  _ _ _|8   5|  _ _ _ _|8

  1| |    1| |      1| |        1| |          1| |

  1| |    1| |      1| |        1| |          1| |

  1| |    1| |      1| |        1| |          1|_|

  1| |    1| |      1| |        1|_|

  1| |    1| |      1|_|

  1| |    1|_|

  1|_|

    _ _ _ _ _ _       _ _ _ _ _ _ _       _ _ _ _ _ _ _ _

  6|  _ _ _ _ _|8   7|  _ _ _ _ _ _|8   8|_ _ _ _ _ _ _ _|8

  1| |              1|_|

  1|_|

.

    _ _       _ _ _       _ _ _ _       _ _ _ _ _       _ _ _ _ _ _

  2|  _|7   3|  _ _|7   4|  _ _ _|7   5|  _ _ _ _|7   6|  _ _ _ _ _|7

  2| |_|1   2| |_|  1   2| |_|    1   2| |_|      1   2|_|_|        1

  1| |      1| |        1| |          1|_|

  1| |      1| |        1|_|

  1| |      1|_|

  1|_|

.

    _ _       _ _ _       _ _ _       _ _ _ _       _ _ _ _       _ _ _ _ _

  2|  _|6   3|  _ _|6   3|  _ _|6   4|  _ _ _|6   4|  _ _ _|6   5|  _ _ _ _|6

  2| | |2   2| | |  2   3| |_ _|2   2| | |    2   3| |_ _|  2   3|_|_ _|    2

  2| |_|    2| |_|      1| |        2|_|_|        1|_|

  1| |      1|_|        1|_|

  1|_|

.

    _ _       _ _ _        _ _ _ _

  2|  _|5   3|  _ _|5    4|  _ _ _|5

  2| | |3   3| |  _|3    4|_|_ _ _|3

  2| | |    2|_|_|

  2|_|_|

.

There are  5 right angles of size 1, so T(8,1) = 5.

There are  6 right angles of size 2, so T(8,2) = 6.

There are  3 right angles of size 3, so T(8,3) = 3.

There are no right angle  of size 4, so T(8,4) = 0.

There are  3 right angles of size 5, so T(8,5) = 3.

There are  6 right angles of size 6, so T(8,6) = 6.

There are  5 right angles of size 7, so T(8,7) = 5.

There are  8 right angles of size 8, so T(8,8) = 8.

Hence the 8th row of triangle is [5, 6, 3, 0, 3, 6, 5, 8].

Note that the sum of the terms after the last zero is 3 + 6 + 5 + 8 = 22, equaling A000041(8) = 22, the number of partitions of 8.

CROSSREFS

Row sums give A115995.

Right border gives A000027.

Cf. A000041, A049597, A083480, A083906, A188674, A259481, A325458, A330375, A330378, A330379.

Sequence in context: A171380 A323592 A170980 * A309577 A029301 A263414

Adjacent sequences:  A330366 A330367 A330368 * A330370 A330371 A330372

KEYWORD

nonn,tabl,more

AUTHOR

Omar E. Pol, Dec 12 2019.

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified July 24 23:25 EDT 2021. Contains 346273 sequences. (Running on oeis4.)