cp's OEIS Frontend

A331638 Number of binary matrices with nonzero rows, a total of n ones and each column with the same number of ones and columns in nonincreasing lexicographic order.

%I A331638 #18 Apr 04 2025 15:38:36
%S A331638 1,3,5,16,17,140,65,1395,2969,22176,1025,1050766,4097,13010328,
%T A331638 128268897,637598438,65537,64864962683,262145,1676258452736,
%U A331638 28683380484257,24908619669860,4194305,30567710172480050,8756434134071649,62128557507554504,21271147396968151093
%N A331638 Number of binary matrices with nonzero rows, a total of n ones and each column with the same number of ones and columns in nonincreasing lexicographic order.
%C A331638 The condition that the columns be in nonincreasing order is equivalent to considering nonequivalent matrices up to permutation of columns.
%C A331638 From _Gus Wiseman_, Apr 03 2025: (Start)
%C A331638 Also the number of multiset partitions such that (1) the blocks together cover an initial interval of positive integers, (2) the blocks are sets of a common size, and (3) the block-sizes sum to n. For example, the a(1) = 1 through a(4) = 16 multiset partitions are:
%C A331638   {{1}}  {{1,2}}    {{1,2,3}}      {{1,2,3,4}}
%C A331638          {{1},{1}}  {{1},{1},{1}}  {{1,2},{1,2}}
%C A331638          {{1},{2}}  {{1},{1},{2}}  {{1,2},{1,3}}
%C A331638                     {{1},{2},{2}}  {{1,2},{2,3}}
%C A331638                     {{1},{2},{3}}  {{1,2},{3,4}}
%C A331638                                    {{1,3},{2,3}}
%C A331638                                    {{1,3},{2,4}}
%C A331638                                    {{1,4},{2,3}}
%C A331638                                    {{1},{1},{1},{1}}
%C A331638                                    {{1},{1},{1},{2}}
%C A331638                                    {{1},{1},{2},{2}}
%C A331638                                    {{1},{1},{2},{3}}
%C A331638                                    {{1},{2},{2},{2}}
%C A331638                                    {{1},{2},{2},{3}}
%C A331638                                    {{1},{2},{3},{3}}
%C A331638                                    {{1},{2},{3},{4}}
%C A331638 (End)
%H A331638 Andrew Howroyd, <a href="/A331638/b331638.txt">Table of n, a(n) for n = 1..200</a>
%F A331638 a(n) = Sum_{d|n} A330942(n/d, d).
%F A331638 a(p) = 2^(p-1) + 1 for prime p.
%Y A331638 Cf. A330942, A331639.
%Y A331638 For constant instead of strict blocks we have A034729.
%Y A331638 Without equal sizes we have A116540 (normal set multipartitions).
%Y A331638 Without strict blocks we have A317583.
%Y A331638 For distinct instead of equal sizes we have A382428, non-strict blocks A326517.
%Y A331638 For equal sums instead of sizes we have A382429, non-strict blocks A326518.
%Y A331638 Normal multiset partitions: A255903, A255906, A317532, A382203, A382204, A382216.
%Y A331638 Cf. A000110, A000670, A007716, A034691, A035310, A050320, A050326, A116539, A306319, A381718.
%K A331638 nonn
%O A331638 1,2
%A A331638 _Andrew Howroyd_, Jan 23 2020