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We calculate cross section for production of $D$ mesons and $\Lambda_c$baryons in proton-proton collisions at the LHC. The cross section forproduction of $c \bar c$ pairs is calculated within $k_T$-factorizationapproach with the Kimber-Martin-Ryskin unintegrated gluon distributionsobtained on the basis of modern collinear gluon distribution functions. We showthat our approach well describes the $D^0$, $D^+$ and $D_s$ experimental data.We try to understand recent ALICE and LHCb data for $\Lambda_c$ production withthe $c \to \Lambda_c$ independent parton fragmentation approach. The Petersonfragmentation functions are used. The $f_{c \to \Lambda_c}$ fragmentationfraction and $\varepsilon_{c}^{\Lambda}$ parameter for $c \to \Lambda_c$ arevaried. As a control plot we show transverse momentum distribution of differentspecies of $D$ mesons assuming standard values of the $f_{c \to D}$fragmentation fractions known from the literature. The fraction $f_{c \to\Lambda_c}$ neccessary to describe the ALICE data is much larger than theaverage value obtained from $e^+ e^-$ or $e p$ experiments. No drasticmodification of the shape of fragmentation function is allowed by the new ALICEand LHCb data for $\Lambda_c$ production. We also discuss a possible dependenceof the $\Lambda_c/ D^0$ baryon-to-meson ratio on rapidity and transversemomentum as seems observed recently by the ALICE and LHCb collaborations. Threedifferent effects are considered: the value of $\varepsilon_c^{\Lambda}$parameter in Peterson fragmentation function for $c \to \Lambda_c$, akinematical effect related to the hadronization prescription and a possiblefeed-down from higher charmed-baryon excitations. It seems very difficult, ifnot impossible, to understand the ALICE data within the considered independentparton fragmentation scheme.

Production of $\Lambda_c$ baryons at the LHC within the $k_T$-factorization approach and independent parton fragmentation picture