On the Intriguing Problem of Counting (n+1,n+2)-Core Partitions into Odd Parts

Tewodros Amdeberhan and Armin Straub initiated the study of enumerating subfamilies of the set of (s,t)-core partitions. While the enumeration of (n+1,n+2)-core partitions into distinct parts is relatively easy (in fact it equals the Fibonacci number F_{n+2}), the enumeration of (n+1,n+2)-core partitions into odd parts remains elusive. Straub computed the first eleven terms of that sequence, and asked for a "formula," or at least a fast way, to compute many terms. While we are unable to find a "fast" algorithm, we did manage to find a "faster" algorithm, which enabled us to compute 23 terms of this intriguing sequence. We strongly believe that this sequence has an algebraic generating function, since a "sister sequence" (see the article), is OEIS sequence A047749 that does have an algebraic generating function. One of us (DZ) is pledging a donation of 100 dollars to the OEIS, in honor of the first person to generate sufficiently many terms to conjecture (and prove non-rigorously) an algebraic equation for the generating function of this sequence, and another 100 dollars for a rigorous proof of that conjecture. Finally, we also develop algorithms that find explicit generating functions for other, more tractable, families of (n+1,n+2)-core partitions.Comment: 12 pages, accompanied by Maple package. This version announces that our questions were all answered by Paul Johnson, and a donation to the OEIS, in his honor, has been mad

Effects of <i>Dittrichia viscosa</i> volatiles on amino acids content.

<p>The effects of the exposition for 12 days to <i>D</i>. <i>viscosa</i> VOCs on lettuce leaf amino acids abundance. Asp (aspartic acid); Glu (glutamic acid); Leu (leucine); Thr (threonine); Val (valine); Ser (serine); GABA (γ-aminobutiric acid); Ile (isoleucine); Pro (proline). Data were analyzed through t-test (P≤0.05) (data from <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0170161#pone.0170161.s002" target="_blank">S1 Table</a>). * <i>P</i> < 0.05; ** <i>P</i> < 0.01; *** <i>P</i> < 0.001. N = 4.</p

Effects of <i>Dittrichia viscosa</i> volatiles on photochemical quantum yield of the PSII, the quantum yield of light-induced nonphotochemical quenching and chlorophyll fluorescence.

<p>Values of the effective photochemical quantum yield of the light adapted PSII <i>Φ</i>_II, the quantum yield of light-induced nonphotochemical quenching <i>Φ</i>_NPQ and chlorophyll fluorescence <i>Φ</i>_NO in whole lettuce plants after <i>D</i>. <i>viscosa</i> VOCs exposition (50 g of plant material). Asterisks along the curves indicate statistical differences with (<i>P</i> ≤ 0.05). * <i>p</i> < 0.05; ** <i>p</i> < 0.01; *** <i>p</i> < 0.001. T₀ –T₄ = days of treatment. AU = Arbitrary Units. N = 3.</p

Effects of <i>Dittrichia viscosa</i> volatiles on sugars content.

<p>The effects of the exposition for 12 days to <i>D</i>. <i>viscosa</i> VOCs on lettuce leaf sugars abundance. Suc (Sucrose); D-Glu (D-Glucose); D-Lac (D-Lactose); Ara (Arabinose); β-Gentiobiose (β-Gen); D-Xyl (D-Xylose); Myo (Myoinositol); Gal (Galactinol). Data were analyzed through t-test (<i>P</i> ≤ 0.05) (data from <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0170161#pone.0170161.s002" target="_blank">S1 Table</a>). * <i>P</i> < 0.05; ** <i>P</i> < 0.01; *** <i>P</i> < 0.001. N = 4.</p

Chemical characterization of <i>Dittrichia viscosa</i> VOCs.

<p><i>Dittrichia viscosa</i> VOCs characterization. Aldehydes: 1–4; alcohols: 5–6; monoterpenes: 7–17, 19, 21–24; homoterpene: 20; sesquiterpenes: 25–39.</p

Effects of <i>Dittrichia viscosa</i> VOCs on several morphological and physiological parameters.

<p>Effects of <i>D</i>. <i>viscosa</i> VOCs on lettuce adult plants. A) Fresh weight (FW); B) dry weight (DW); C) DW/FW ratio; D) relative water content (RWC); E) leaf osmotic potential [Ψ(π)]; F) membrane stability index (MSI); G) lipid peroxidation (MDA) (nmol/mL/g_DW); H) total protein content (μg of protein /g DW). Data are given in percentage compared to the control and were analyzed through LSD test. (<i>P</i> < 0.05). * <i>P</i> < 0.05; ** <i>P</i> < 0.01; *** <i>P</i> < 0.001. N = 4.</p

Semiquantitative determination of H₂O₂ in plants treated with <i>Dittrichia viscosa</i> volatiles.

<p>Lettuce leaves exposed to <i>D</i>. <i>viscosa</i> VOCs for 12 days showing the localization of the hydrogen peroxide on leaf surface after DAB staining: A) Control leaf; B) treated leaf; C) percentage of the Integrated Optical Density (IOD) obtained through image analysis carried on the <i>in situ</i> semi-quantitative determination of H₂O₂. In dark grey is reported the unaffected (Unaff) area of the leaf, whereas in bright grey the leaf surface interested by H₂O₂ accumulation (Aff). The area affected is expressed as percentage of the total area. (<i>P</i> < 0.05). * <i>P</i> < 0.05; ** <i>P</i> < 0.01; *** <i>P</i> < 0.001. N = 4.</p

PCA analysis carried on the metabolite identified and quantified after <i>Dittrichia viscosa</i> VOCs treatment.

<p><b>A)</b> Principal Component Analysis model scores <b>A)</b> and loading plot <b>B)</b> of metabolite profile of control plants (Contr_1 –Contr_4, replicates of control samples) and plants exposed to <i>D</i>. <i>viscosa</i> VOCs (Treated_1 –Treated_4, replicates of the treated samples). Both score and loading plots were generated using the first two PCs, PC1 <i>vs</i> PC2, with the explained variances shown in brackets; <b>C)</b> Overlay heat map of metabolite profiles in plants exposed to <i>D</i>. <i>viscosa</i> VOCs released by fresh aerial parts in comparison with control plants. Each square represents the effect of plant VOCs on the amount of every metabolite using a false-color scale. Red or green regions indicate increased or decreased metabolite content, respectively.</p

Effects of <i>Dittrichia viscosa</i> VOCs on maximum quantum efficiency of dark-adapted PSII and apparent electron transport rate.

<p>Values of maximum quantum efficiency of dark-adapted PSII (<i>Fv/Fm</i>) and apparent electron transport rate (<i>ETR</i>) in whole lettuce plants after <i>D</i>. <i>viscosa</i> VOCs exposition (50 g of plant material). Asterisks along the curves indicate statistical differences with (<i>P</i> ≤ 0.05). * <i>p</i> < 0.05; ** <i>p</i> < 0.01; *** <i>p</i> < 0.001. T₀ –T₄ = days of treatment. AU = Arbitrary Units. N = 3.</p

Effects of <i>Dittrichia viscosa</i> volatiles on germination and root growth.

<p>Effects of <i>D</i>. <i>viscosa</i> VOCs on Total Germination [G_T (%)], Speed Germination (S) and Total Root Length [TRL (cm)] of <i>L</i>. <i>sativa</i>. The nonlinear regression fitting of all the dose-response curves pointed out a significance level of <i>P</i> < 0.001. Different letters along the curve indicate statistical differences with <i>P</i> ≤ 0.05 (LSD). AU = arbitrary units. N = 4.</p