Building on a bijection of Vandervelde, we enumerate certain unimodal sequences whose alternating sum equals zero. This enables us to refine the enumeration of strict partitions with respect to the number of parts and the BG-rank.
Inspired by Andrews 2-colored generalized Frobenius partitions, we consider certain weighted 7-colored partition functions and establish some interesting Ramanujan-type identities and congruences. Moreover, we provide combinatorial interpretations of
some congruences modulo 5 and 7. Finally, we study the properties of weighted 7-colored partitions weighted by the parity of certain partition statistics.
We present combinatorial and analytical results concerning a Sheffer sequence with a generating function of the form $G(x,z)=Q(z)^{x}Q(-z)^{1-x}$, where $Q$ is a quadratic polynomial with real zeros. By using the properties of Riordan matrices we add
ress combinatorial properties and interpretations of our Sheffer sequence of polynomials and their coefficients. We also show that apart from two exceptional zeros, the zeros of polynomials with large enough degree in such a Sheffer sequence lie on the line $x=1/2+it$.
We consider $m$-divisible non-crossing partitions of ${1,2,ldots,mn}$ with the property that for some $tleq n$ no block contains more than one of the first $t$ integers. We give a closed formula for the number of multi-chains of such non-crossing par
titions with prescribed number of blocks. Building on this result, we compute Chapotons $M$-triangle in this setting and conjecture a combinatorial interpretation for the $H$-triangle. This conjecture is proved for $m=1$.
In this paper we demonstrate connections between three seemingly unrelated concepts. (1) The discrete isoperimetric problem in the infinite binary tree with all the leaves at the same level, $ {mathcal T}_{infty}$: The $n$-th edge isoperimetric n
umber $delta(n)$ is defined to be $min_{|S|=n, S subset V({mathcal T}_{infty})} |(S,bar{S})|$, where $(S,bar{S})$ is the set of edges in the cut defined by $S$. (2) Signed almost binary partitions: This is the special case of the coin-changing problem where the coins are drawn from the set ${pm (2^d - 1): $d$ is a positive integer}$. The quantity of interest is $tau(n)$, the minimum number of coins necessary to make change for $n$ cents. (3) Certain Meta-Fibonacci sequences: The Tanny sequence is defined by $T(n)=T(n{-}1{-}T(n{-}1))+T(n{-}2{-}T(n{-}2))$ and the Conolly sequence is defined by $C(n)=C(n{-}C(n{-}1))+C(n{-}1{-}C(n{-}2))$, where the initial conditions are $T(1) = C(1) = T(2) = C(2) = 1$. These are well-known meta-Fibonacci sequences. The main result that ties these three together is the following: $$ delta(n) = tau(n) = n+ 2 + 2 min_{1 le k le n} (C(k) - T(n-k) - k).$$ Apart from this, we prove several other results which bring out the interconnections between the above three concepts.
In this work, we give combinatorial proofs for generating functions of two problems, i.e., flushed partitions and concave compositions of even length. We also give combinatorial interpretation of one problem posed by Sylvester involving flushed parti
tions and then prove it. For these purposes, we first describe an involution and use it to prove core identities. Using this involution with modifications, we prove several problems of different nature, including Andrews partition identities involving initial repetitions and partition theoretical interpretations of three mock theta functions of third order $f(q)$, $phi(q)$ and $psi(q)$. An identity of Ramanujan is proved combinatorially. Several new identities are also established.