tokens: of star formation, with the younger and older parent stellar population dominating the brighter and fainter PNe respectively. Indeed Valenzuela et al. (2019) showed theoretically that the shape of the PNLF at the faint-end is quite sensitive to the distribution of masses of the PN central stars, which is a product of the SFH in any galaxy. In Bhattacharya et al. (2019b , hereafter Paper I), we observed a rise in the faint-end of the PNLF in the central 16 sq. deg. of M 31 conceivably linked to the SFH of M 31. Owing to the homogeneous spatial coverage and photometry of the current imaging survey, we detect PNe associated with all the stellar populations in a 54 sq. deg area centred on M31. Thanks to the sensitivity and accuracy, we can quantify the imprints of the variations of these stellar populations on the PNLFs in distinct regions. In Sect. 2 , we describe our PN survey of M 31 and set the geometrical boundaries around any regions with RGB overdensities where the PNLFs are measured. In Sect. 3 , we obtain the PNLFs, compute the parameters that best describe them, and carry out the quantitative comparisons via statistical methods. In Sect. 4 , we correlate the stellar population parameters with the properties of the PNLFs at both their bright and faint ends in the different subregions. In Sect. 5 , we discuss the implications of our results on the origin of the M31 inner halo substructures. We draw our conclusions in Sect. 6 . 2. Survey of Planetary Nebulae in M31 2.1. Survey characteristics In Paper I, we identified PNe candidates in a 16 sq. deg. uniform imaging survey of the disc and disc–halo interface of M 31 using the MegaCam wide-field imager ( Boulade et al. 2003 ) mounted on the 3.6-meter Canada-France-Hawaii Telescope (CFHT). M 31 was observed through a narrow-band [O III ] filter ( λ c = 5007 Å, Δ λ = 102 Å, on-band) and a broad-band g -filter ( λ c = 4750 Å, Δ λ = 1540 Å, off-band). An additional 38 sq. deg. were obtained in late-2019, providing uniform coverage of the inner halo of M 31 and its substructures reaching a uniform sensitivity down to fluxes equivalent to m 5007 = 26.2 mag over the entire area. Details of these observations are presented in Bhattacharya (2020) and Bhattacharya et al. (in prep.). The zero points for each of the [O III ] and g -band frames in AB magnitudes, normalised to a 1 s exposure, are Z [O III ] = 23.4 and Z g = 26.5. The photometry was calibrated using observations of spectrophotometric standard stars. The photometric uncertainty (err m , 5007 ) is well below 0.01 mag at m 5007 = 22.5 mag; it increases to err m , 5007 ∼ 0.15–0.2 mag at m 5007 = 26.2 mag. The PN identification in each of these pointings was also carried out as described in Paper I using the on-off band technique developed and validated in Arnaboldi et al. (2002 , 2003) , later optimised for large imaging surveys by Longobardi et al. (2013) . The PNe are identified as point-like sources with an excess [O III ] – g colour. The
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