cp's OEIS Frontend

A multiset partition of weight n is a finite multiset of finite nonempty multisets whose sizes sum to n.

Number of distinct nonnegative integer matrices with all row sums equal and total sum n up to row and column permutations. - Andrew Howroyd, Sep 05 2018

From Gus Wiseman, Oct 11 2018: (Start)

Also the number of non-isomorphic multiset partitions of weight n in which each vertex appears the same number of times. For n = 4, non-isomorphic representatives of these 17 multiset partitions are:

{{1,1,1,1}}

{{1,1,2,2}}

{{1,2,3,4}}

{{1},{1,1,1}}

{{1},{1,2,2}}

{{1},{2,3,4}}

{{1,1},{1,1}}

{{1,1},{2,2}}

{{1,2},{1,2}}

{{1,2},{3,4}}

{{1},{1},{1,1}}

{{1},{1},{2,2}}

{{1},{2},{1,2}}

{{1},{2},{3,4}}

{{1},{1},{1},{1}}

{{1},{1},{2},{2}}

{{1},{2},{3},{4}}

(End)

Also the number of labeled hypergraphs spanning an initial interval of positive integers with edge-sizes summing to n. - Gus Wiseman, Dec 18 2018

Sequence corresponds also to the largest number that may be determined by asking no more than 2^(n-1) - 1 questions("Yes"-or-"No" answerable) with lying allowed at most once. - Lekraj Beedassy, Jul 15 2002

Number of Ouroborean rings for binary n-tuplets. - Lekraj Beedassy, May 06 2004

Also the number of games of Nim that are wins for the second player when the starting position is either the empty heap or heaps of sizes 1 <= t_1 < t_2 < ... < t_k < 2^(n-1) (if n is 1, the only starting position is the empty heap). E.g.: a(4) = 16: the winning positions for the second player when all the heap sizes are different and less than 2^3: (4,5,6,7), (3,5,6), (3,4,7), (2,5,7), (2,4,6), (2,3,6,7), (2,3,4,5), (1,6,7), (1,4,5), (1,3,5,7), (1,3,4,6), (1,2,5,6), (1,2,4,7), (1,2,3), (1,2,3,4,5,6,7) and the empty heap. - Kennan Shelton (kennan.shelton(AT)gmail.com), Apr 14 2006

a(n + 1) = 2^(2^n-n-1) = 2^A000295(n) is also the number of set-systems on n vertices with no singletons. The case with singletons is A058891. The unlabeled case is A317794. The spanning/covering case is A323816. The antichain case is A006126. The connected case is A323817. The uniform case is A306021(n) - 1. The graphical case is A006125. The chain case is A005840. - Gus Wiseman, Feb 01 2019

Named after the Dutch mathematician Nicolaas Govert de Bruijn (1918-2012). - Amiram Eldar, Jun 20 2021

The Qian reference has one incorrect term. The formula given in corollary 2.6 also contains a minor error. The second summation needs to be over p_i*p_j*p_h/lcm(p_i, p_j, p_h) rather than gcd(p_i, p_j, p_h)^2. - Andrew Howroyd, Dec 11 2018

A multiset is normal if it spans an initial interval of positive integers.

a(n) is the number of nonnegative integer matrices with total sum n, nonzero rows and each column with the same sum with columns in nonincreasing lexicographic order. - Andrew Howroyd, Jan 15 2020

We define a hypergraph to be any finite set of finite nonempty sets. A hypergraph is uniform if all edges have the same size, and regular if all vertices have the same degree. The span of a hypergraph is the union of its edges.

Also the number of 0-1 matrices with n columns, all distinct rows, no zero columns, equal row-sums, and equal column-sums, up to a permutation of the rows.

Also the number of uniform multiset partitions of the multiset of prime indices of n, where a multiset partition is uniform if all parts have the same size.

The common average is necessarily (n+1)/2. The number of blocks with this average is given by A070925. - Christian Sievers, Aug 22 2024