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A072233 Square array T(n,k) read by antidiagonals giving number of ways to distribute n indistinguishable objects in k indistinguishable containers; containers may be left empty. 50
1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 2, 2, 1, 1, 0, 1, 3, 3, 2, 1, 1, 0, 1, 3, 4, 3, 2, 1, 1, 0, 1, 4, 5, 5, 3, 2, 1, 1, 0, 1, 4, 7, 6, 5, 3, 2, 1, 1, 0, 1, 5, 8, 9, 7, 5, 3, 2, 1, 1, 0, 1, 5, 10, 11, 10, 7, 5, 3, 2, 1, 1, 0, 1, 6, 12, 15, 13, 11, 7, 5, 3, 2, 1, 1, 0, 1, 6, 14, 18, 18, 14, 11, 7, 5, 3, 2, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,13

COMMENTS

Regarded as a triangular table, this is another version of the number of partitions of n into k parts, A008284. - Franklin T. Adams-Watters, Dec 18 2006

From Gus Wiseman, Feb 10 2021: (Start)

T(n,k) is also the number of partitions of n with greatest part k, if we assume the greatest part of an empty partition to be 0. Row n = 9 counts the following partitions:

  111111111  22221     333      432     54     63    72   81  9

             222111    3222     441     522    621   711

             2211111   3321     4221    531    6111

             21111111  32211    4311    5211

                       33111    42111   51111

                       321111   411111

                       3111111

(End)

LINKS

Robert G. Wilson v, Table of n, a(n) for n = 0..10010

Combinatorial Object Server, Information on Numerical Partitions

FindStat - Combinatorial Statistic Finder, The length of the partition.

FORMULA

T(0, k) = 1, T(n, 0) = 0 (n>0), T(1, k) = 1 (k>0), T(n, 1) = 1 (n>0), T(n, k) = 0 for n < 0, T(n, k) = Sum[ T(n-k+i, k-i), i=0...k-1] Or, T(n, 1) = T(n, n) = 1, T(n, k) = 0 (k>n), T(n, k) = T(n-1, k-1) + T(n-k, k).

G.f. Product_{j=0..infinity} 1/(1-xy^j). Regarded as a triangular array, g.f. Product_{j=1..infinity} 1/(1-xy^j). - Franklin T. Adams-Watters, Dec 18 2006

O.g.f. of column No. k of the triangle a(n,k) is x^k/product(1-x^j,j=1..k), k>=0 (the undefined product for k=0 is put to 1). - Wolfdieter Lang, Dec 03 2012

EXAMPLE

Table begins (upper left corner = T(0,0)):

1 1 1  1  1  1  1  1  1 ...

0 1 1  1  1  1  1  1  1 ...

0 1 2  2  2  2  2  2  2 ...

0 1 2  3  3  3  3  3  3 ...

0 1 3  4  5  5  5  5  5 ...

0 1 3  5  6  7  7  7  7 ...

0 1 4  7  9 10 11 11 11 ...

0 1 4  8 11 13 14 15 15 ...

0 1 5 10 15 18 20 21 22 ...

There is 1 way to distribute 0 objects into k containers: T(0, k) = 1. The different ways for n=4, k=3 are: (oooo)()(), (ooo)(o)(), (oo)(oo)(), (oo)(o)(o), so T(4, 3) = 4.

From Wolfdieter Lang, Dec 03 2012 (Start)

The triangle a(n,k) = T(n-k,k) begins:

n\k  0  1  2  3  4  5  6  7  8  9 10 ...

00   1

01   0  1

02   0  1  1

03   0  1  1  1

04   0  1  2  1  1

05   0  1  2  2  1  1

06   0  1  3  3  2  1  1

07   0  1  3  4  3  2  1  1

08   0  1  4  5  5  3  2  1  1

09   0  1  4  7  6  5  3  2  1  1

10   0  1  5  8  9  7  5  3  2  1  1

...

Row n=5 is, for k=1..5, [1,2,2,1,1] which gives the number of partitions of n=5 with k parts. See A008284 and the Franklin T. Adams-Watters comment above. (End)

From Gus Wiseman, Feb 10 2021: (Start)

Row n = 9 counts the following partitions:

  9  54  333  3222  22221  222111  2211111  21111111  111111111

     63  432  3321  32211  321111  3111111

     72  441  4221  33111  411111

     81  522  4311  42111

         531  5211  51111

         621  6111

         711

(End)

MATHEMATICA

Flatten[Table[Length[IntegerPartitions[n, {k}]], {n, 0, 20}, {k, 0, n}]] (* Emanuele Munarini, Feb 24 2014 *)

PROG

(Sage)

from sage.combinat.partition import number_of_partitions_length

[[number_of_partitions_length(n, k) for k in (0..n)] for n in (0..10)] # Peter Luschny, Aug 01 2015

CROSSREFS

Sum of antidiagonal entries T(n, k) with n+k=m equals A000041(m).

Alternating row sums are A081362.

Cf. A008284.

The version for factorizations is A316439.

The version for set partitions is A048993/A080510.

The version for strict partitions is A008289/A059607.

A047993 counts balanced partitions, ranked by A106529.

A063995/A105806 count partitions by Dyson rank.

Cf. A006141, A064174, A096401, A117409, A168659, A215366.

Sequence in context: A344612 A069713 A319453 * A264391 A116598 A244925

Adjacent sequences:  A072230 A072231 A072232 * A072234 A072235 A072236

KEYWORD

easy,nonn,tabl

AUTHOR

Martin Wohlgemuth (mail(AT)matroid.com), Jul 05 2002

EXTENSIONS

Corrected by Franklin T. Adams-Watters, Dec 18 2006

STATUS

approved

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Last modified May 26 11:30 EDT 2022. Contains 354086 sequences. (Running on oeis4.)