Regularly spaced subsums of integer partitions

For integer partitions $\lambda :n=a_1+...+a_k$, where $a_1\ge a_2\ge >...\ge a_k\ge 1$, we study the sum $a_1+a_3+...$ of the parts of odd index. We show that the average of this sum, over all partitions $\lambda$ of $n$, is of the form $n/2+(\sqrt{6}/(8\pi))\sqrt{n}\log{n}+c_{2,1}\sqrt{n}+O(\log{n}).$ More generally, we study the sum $a_i+a_{m+i}+a_{2m+i}+...$ of the parts whose indices lie in a given arithmetic progression and we show that the average of this sum, over all partitions of $n$, is of the form $n/m+b_{m,i}\sqrt{n}\log{n}+c_{m,i}\sqrt{n}+O(\log{n})$, with explicitly given constants $b_{m,i},c_{m,i}$. Interestingly, for $m$ odd and $i=(m+1)/2$ we have $b_{m,i}=0$, so in this case the error term is of lower order. The methods used involve asymptotic formulas for the behavior of Lambert series and the Zeta function of Hurwitz. We also show that if $f(n,j)$ is the number of partitions of $n$ the sum of whose parts of even index is $j$, then for every $n$, $f(n,j)$ agrees with a certain universal sequence, Sloane's sequence \texttt{#A000712}, for $j\le n/3$ but not for any larger $j$

Results on Normal Forms for FPU Chains

In this paper we prove, among other results, that near the equilibirum position, any periodic FPU chain with an odd number N of particles admits a Birkhoff normal form up to order 4, whereas any periodic FPU chain with N even admits a resonant normal form up to order 4. This resonant normal form of order 4 turns out to be completely integrable. Further, for N odd, we obtain an explicit formula of the Hessian of its Hamiltonian at the fixed poin

Restriction of Odd Degree Characters of Sn\mathfrak{S}_n

Let $n$ and $k$ be natural numbers such that $2^k < n$. We study the restriction to $\mathfrak{S}_{n-2^k}$ of odd-degree irreducible characters of the symmetric group $\mathfrak{S}_n$. This analysis completes the study begun in [Ayyer A., Prasad A., Spallone S., Sem. Lothar. Combin. 75 (2015), Art. B75g, 13 pages] and recently developed in [Isaacs I.M., Navarro G., Olsson J.B., Tiep P.H., J. Algebra 478 (2017), 271-282]

Higher Order SPT-Functions

Andrews' spt-function can be written as the difference between the second symmetrized crank and rank moment functions. Using the machinery of Bailey pairs a combinatorial interpretation is given for the difference between higher order symmetrized crank and rank moment functions. This implies an inequality between crank and rank moments that was only know previously for sufficiently large n and fixed order. This combinatorial interpretation is in terms of a weighted sum of partitions. A number of congruences for higher order spt-functions are derived.Comment: 21 pages (previous version was 19 pages), added reference to Andrews and Rose's recent paper, MacMahon's paper and OEIS, changed some wordin