[seqfan] a possibly significant coincidence on A003509

Wouter Meeussen wouter.meeussen at telenet.be
Fri Aug 11 11:45:28 CEST 2017

A003509 : Let k(m) denote the least integer such that every m X m 
(0,1)-matrix with exactly k(m) ones in each row and in each column contains 
a 2 X 2 submatrix without zeros. The sequence gives the index n of the first 
term in each string of equal entries in the {k(m)} sequence (see A155934).
(Formerly M0833)

A003509 =  2, 3, 7, 13, 21, 31

by coincidence, when looking at the rows of the upper-triangular 
Kostka-matrix, I found that the count of rows (each indexed by a partition) 
that contain one or more zero's, as function of the size n of the  P(n)xP(n) 
Kostka matrix, equals :

0, 0, 0, 0, 0, 2, 3, 7, 13, 21, 31, 50, 70, 99, 137, 188, 250, 334, 433, 566

Since both sequences loosely concern counting zero's, there could possibly 
be some connection.
The five leading zero's are a severe counter-indication however.

Probably just an other example of the strong law of small integers.


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