Tewodros Amdeberhan

In this paper, we investigate the arithmetic properties of the difference between the number of partitions of a positive integer n with even crank and those with odd crank, denoted $$C(n) = c_e(n) - c_o(n).$$ C ( n ) = c e ( n ) - c o ( n ) . Inspired by Ramanujan’s classical congruences for the partition function p ( n ), we establish a Ramanujan-type congruence for C ( n ), proving that $$C(5n+4) \equiv 0 \pmod {5}.$$ C ( 5 n + 4 ) ≡ 0 ( mod 5 ) . Further, we study the generating function $$\sum _{n=0}^\infty a(n)\, q^n = \frac{(-q; q)^2_\infty }{(q; q)_\infty },$$ ∑ n = 0 ∞ a ( n ) q n = ( - q ; q ) ∞ 2 ( q ; q ) ∞ , which arises naturally in this context, and provide multiple combinatorial interpretations for the sequence a ( n ). We then offer a complete characterization of the values $$a(n) \mod 2^m$$ a ( n ) mod 2 m for $$m = 1, 2, 3, 4,$$ m = 1 , 2 , 3 , 4 , highlighting their connection to generalized pentagonal numbers. Using computational methods and modular forms, we also derive new identities and congruences, including $$a(7n+2) \equiv 0 \pmod {7},$$ a ( 7 n + 2 ) ≡ 0 ( mod 7 ) , expanding the scope of partition congruences in arithmetic progressions. These results build upon classical techniques and recent computational advances, revealing deep combinatorial and modular structure within partition functions.

In a fascinating recent American Mathematical Monthly article, Norman Wildberger and Dean Rubine introduced a new kind of combinatorial numbers, that they aptly named the ``Geode numbers''. While their definition is simple, these numbers are surprisingly hard to compute, in general. While the two-dimensional case has a nice closed-form expression, that make them easy to compute, already the three-dimensional case poses major computational challenges that we do meet, combining experimental mathematics and the holonomic ansatz. Alas, things get really complicated in four and higher dimensions, and we are unable to efficiently compute, for example, the $1000$-th term of the four-dimensional diagonal Geode sequence. A donation of $100$ US dollars to the OEIS, in honor of the first person to compute this number, is offered.

In this paper, we investigate the arithmetic properties of the difference between the number of partitions of a positive integer $n$ with even crank and those with odd crank, denoted $C(n)=c_e(n)-c_o(n)$. Inspired by Ramanujan's classical congruences for the partition function $p(n)$, we establish a Ramanujan-type congruence for $C(n)$, proving that $C(5n+4) \equiv 0 \pmod{5}$. Further, we study the generating function $\sum\limits_{n=0}^\infty a(n)\, q^n = \frac{(-q; q)^2_\infty}{(q; q)_\infty}$, which arises naturally in this context, and provide multiple combinatorial interpretations for the sequence $a(n)$. We then offer a complete characterization of the values $a(n) \mod 2^m$ for $m = 1, 2, 3, 4$, highlighting their connection to generalized pentagonal numbers. Using computational methods and modular forms, we also derive new identities and congruences, including $a(7n+2) \equiv 0 \pmod{7}$, expanding the scope of partition congruences in arithmetic progressions. These results build upon classical techniques and recent computational advances, revealing deep combinatorial and modular structure within partition functions.

We prove an explicit formula for the $p$-adic valuation of the Legendre polynomials $P_n(x)$ evaluated at a prime $p$, and generalize an old conjecture of the third author. We also solve a problem proposed by Cigler in 2017.

In his “lost notebook,” Ramanujan used iterated derivatives of two theta functions to define sequences of q-series $\{U_{2t}(q)\}$ and $\{V_{2t}(q)\}$ that he claimed to be quasimodular. We give the first explicit proof of this claim by expressing them in terms of “partition Eisenstein series,” extensions of the classical Eisenstein series $E_{2k}(q),$ defined by $$ \begin{align*}\lambda=(1^{m_1}, 2^{m_2},\dots, n^{m_n}) \vdash n \ \ \ \longmapsto \ \ \ E_{\lambda}(q):= E_2(q)^{m_1} E_4(q)^{m_2}\cdots E_{2n}(q)^{m_n}. \end{align*} $$ For functions $\phi : \mathcal {P}\mapsto {\mathbb C}$ on partitions, the weight $2n$ partition Eisenstein trace is $$ \begin{align*}\operatorname{\mathrm{Tr}}_n(\phi;q):=\sum_{\lambda \vdash n} \phi(\lambda)E_{\lambda}(q). \end{align*} $$ For all t, we prove that $U_{2t}(q)=\operatorname {\mathrm {Tr}}_t(\phi _U;q)$ and $V_{2t}(q)=\operatorname {\mathrm {Tr}}_t(\phi _V;q),$ where $\phi _U$ and $\phi _V$ are natural partition weights, giving the first explicit quasimodular formulas for these series.