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Consider the transition matrix $R = \left(R_{\lambda,\mu}\right)$, defined by $$ p_\lambda = \sum_{\mu} R_{\lambda\mu}m_\mu , $$ between the power-sum and the monomial basis of the ring of symmetric functions. There are plenty of combinatorial descriptions of $R_{\lambda\mu}$, it is for example the number of ordered brick-tableaux of shape $\lambda$ and brick-sizes determined by $\mu$, see e.g. this excellent paper with transition matrices by Eğecioğlu and Remmel.

In the paper on Bottom Schur functions by P. Clifford and R. Stanley (2004), the authors consider the dimension of the null-space of $R-D$, where $D$ is the diagonal of $R$.

As the size $n$ of the partitions increase, they computed (by computer I presume) the sequence of dimensions $d(n)$ is $$ 1 , 1 , 1 , 2 , 2 , 3 , 4 , 5 , 7 , 9 , 11 , 15 , 19 , 24 $$ but they do not make any conjecture of what this sequence of integers might be.

However, a quick search in OEIS gives one result, A129528, entered in 2007, after Clifford and Stanley's paper was published. The sequence A129528, $a(n)$ is defined as follows: Consider the set of all Dyck paths, consisting of steps $(1,1)$ and $(1,-1)$. The peak-abscissa-sum of a path is the sum of the $x$-coordinates of all peaks. Then $a(n)$ is the number of Dyck paths of any length with peak-abscissa-sum $n$.

Question: Is there some neat way to show that A129528 is at least a lower bound of $d(n)$, by explicitly constructing $a(n)$ linearly independent vectors in the null-space?

A good start is to examine the Bottom Schur functions defined in the paper, as they can be identified with elements in the null space of $R-D$.

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  • $\begingroup$ Have you looked at the other eigenvalues of $RD^{-1}$ ? I wouldn't be surprised if they are integers, and their multiplicities count Dyck paths with other sums of peaks. (Sorry for backseat problemsolving -- I'm currently travelling and have a backlog of lecture notes to write.) $\endgroup$
    – darij grinberg
    Commented Mar 21, 2019 at 0:59
  • $\begingroup$ @darijgrinberg: I don't understand, $R$ is triangular, so $R-D$ has all eigenvalues equal to $0$, no? $\endgroup$
    – FindStat
    Commented Mar 21, 2019 at 20:10
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    $\begingroup$ Oh! You're right. Then I guess it's worth looking at the sizes of the Jordan blocks. $\endgroup$
    – darij grinberg
    Commented Mar 21, 2019 at 21:24
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    $\begingroup$ Here are the block sizes (the last being a partition of length 24): $[1]$, $[2]$, $[3]$, $[4, 1]$, $[5, 2]$, $[6, 3, 2]$, $[7, 4, 3, 1]$, $[8, 5, 4, 3, 2]$, $[9, 6, 5, 4, 3, 2, 1]$, $[10, 7, 6, 5, 5, 3, 3, 2, 1]$, $[11, 8, 7, 6, 6, 4, 4, 3, 3, 2, 2]$, $[12, 9, 8, 7, 7, 6, 5, 4, 4, 4, 4, 3, 2, 1, 1]$, $[13, 10, 9, 8, 8, 7, 6, 5, 5, 5, 5, 4, 4, 4, 2, 2, 2, 1, 1]$, $[14, 11, 10, 9, 9, 8, 8, 6, 6, 6, 6, 5, 5, 5, 5, 4, 3, 3, 3, 3, 2, 2, 1, 1]$ $\endgroup$
    – Martin Rubey
    Commented Mar 22, 2019 at 8:45
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    $\begingroup$ It seems that, whenever there is only one Dyck path with given peak-abscissa-sum, then there is a vector with first non-zero coefficient indexed by the integer partition corresponding to the positions of the peaks. For example, for $n=8$, the kernel is spanned by the vectors $(8)$, $(71)-(44)$, $(62)-(44)$, $(611)-2(521)+(332)$, $(53)-(44)$. There are five Dyck paths with peak-abscissa-sum $8$, and the positions of the peaks are: $(8)$, $(71)$, $(62)$, $(62)$ and $(53)$. $\endgroup$
    – Martin Rubey
    Commented Mar 22, 2019 at 16:36

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