Split columns base on conditions in Python

try:

df['SizeAgeQuantity'] = df['SizeAgeQuantity'].str.split()

size_set = ["4 1/2","7", "9 5/8", "13 3/8", "18", "30", "32"]

def f_size(x):
    if (len(x)==4) and ('/' in x[1]):
        return ' '.join(x[:2])
    elif (len(x)==3):
        return x[0]

    #_______________________________________________________

    elif (len(x)==2) and (x[0] in size_set): #[size, ...]
        return x[0] 
    elif (len(x)==2) and (x[1] in size_set): #[..., size]
        return x[1] 
    elif (len(x)==1) and (x[0] in size_set): #[size]
        return x[0] 
    #_______________________________________________________
    elif (len(x)==2) and (x[0] not in size_set): #[age, qty]
        return None
    elif (len(x)==1) and (x[0] not in size_set): #[age]
        return None
    #_______________________________________________________

    else:
        return "unknown"

#when we don't have size(the values from the set) and there is only one number we will assume it is for age

def f_age(x):
    if (len(x)==4) and ('/' in x[1]):
        return x[2]
    elif (len(x)==3):
        return x[1]
    #_____________________________________________________
    elif (len(x)==2) and (x[0] in size_set): #[size, ...]
        return "unknown"
    elif (len(x)==2) and (x[1] in size_set): #[..., size]
        return "unknown"
    elif (len(x)==1) and (x[0] in size_set): #[size]
        return None
    #_______________________________________________________
    elif (len(x)==2) and (x[0] not in size_set): #[age, qty]
        return x[0]
    elif (len(x)==1) and (x[0] not in size_set): #[age]
        return x[0]
    #_______________________________________________________
    else:
        return "unknown"
def f_qty(x):
    if (len(x)==4) and ('/' in x[1]):
        return x[-1]
    elif (len(x)==3):
        return x[-1]
    #_______________________________________________________

    elif (len(x)==2) and (x[0] in size_set): #[size, ...]
        return "unknown"
    elif (len(x)==2) and (x[1] in size_set): #[..., size]
        return "unknown"
    elif (len(x)==1) and (x[0] in size_set): #[size]
        return None
    #_______________________________________________________

    elif (len(x)==2) and (x[0] not in size_set): #[age, qty]
        return x[1]
    elif (len(x)==1) and (x[0] not in size_set): #[age]
        return None

    
    else:
        return "unknown"

df['size'] = df['SizeAgeQuantity'].map(f_size)
df['age'] = df['SizeAgeQuantity'].map(f_age)
df['qty'] = df['SizeAgeQuantity'].map(f_qty)

df
    Date        Item    SizeAgeQuantity     size    age     qty
0   2-18-2019   NY01    [13, 3/8, 5, 846]   13 3/8  5       846
1   2-18-2019   Ld01    [4, 1/2, 557, 85]   4 1/2   557     85
2   2-19-2019   Du02    [9, 5/8, 47, 4464]  9 5/8   47      4464
3   2-19-2019   Du01    [30, 58]            30      unknown unknown
4   2-20-2019   Ps55    [32, 304, 304]      32      304     304
5   2-21-2019   L55     [7, 6588, 685]      7       6588    685
6   2-21-2019   Du85    [4118, 587]         None    4118    587
7   2-22-2019   L85     [29]                None    29      None

This should do the trick:

import pandas as pd
import numpy as np

df = pd.DataFrame({
    'Date':['2-18-2019','2-18-2019','2-19-2019','2-19-2019','2-20-2019','2-21-2019','2-21-2019','2-22-2019'],
    'Item':['NY01','Ld01','Du02','Du01','Ps55','L55','Du85','L85'],
    'SizeAgeQuantity':['13 3/8 5 846','4 1/2 557 85','9 5/8 47 4464','30 58','32 304 304','7 6588 685','4118 587','29']})

size_list = ["4 1/2", "7", "9 5/8", "13 3/8", "18", "30", "32"]

columns = []

for row in df.SizeAgeQuantity:
    values = row.split()
    # if there's a "/" in the row, 
    # combine values 1 and 2
    if "/" in row:
        size = " ".join(values[:2])
        del values[0:2]
        values.insert(0, size)
    
    # add nan padding to the values list
    values = values + [np.nan] * (3-len(values))
    
    # if the 1st value is not size, shift list right
    if values[0] not in size_list:
        values = values[-1:] + values[:-1]
    
    columns.append(values)

saq = pd.DataFrame(columns, columns=["Size", "Age", "Quantity"])
out = pd.concat([df.drop("SizeAgeQuantity", axis=1), saq], axis=1)
print(out)

Out:

        Date  Item    Size   Age Quantity
0  2-18-2019  NY01  13 3/8     5      846
1  2-18-2019  Ld01   4 1/2   557       85
2  2-19-2019  Du02   9 5/8    47     4464
3  2-19-2019  Du01      30    58      NaN
4  2-20-2019  Ps55      32   304      304
5  2-21-2019   L55       7  6588      685
6  2-21-2019  Du85     NaN  4118      587
7  2-22-2019   L85     NaN    29      NaN

1. Looking at the required result, it is clear that the lines must be broken from the end, since there are no fractional numbers. It is necessary to limit the amount of breakdown by two, since we have the maximum of 3 elements. While we leave the received column in the form of lists. And as a result of a line with three elements, they are ready immediately!
2. With two and one element you have to tinker. We check for the length of the list, and for a match between the beginning of the string and the tuple (not a list!, otherwise the function does not work, and be sure to insert spaces into the strings of the tuple to clearly identify the element). We supplement each list up to three, from the right side, with empty lines.
3. Now we break the "normalized" column into the necessary columns and delete the ones that are no longer needed.

Result:

import pandas as pd

df = pd.DataFrame({
    'Date': ['2-18-2019', '2-18-2019', '2-19-2019', '2-19-2019', '2-20-2019', '2-21-2019', '2-21-2019', '2-22-2019'],
    'Item': ['NY01', 'Ld01', 'Du02', 'Du01', 'Ps55', 'L55', 'Du85', 'L85'],
    'SizeAgeQuantity': ['13 3/8 5 846', '4 1/2 557 85', '9 5/8 47 4464', '30 58', '32 304 304', '7 6588 685',
                        '4118 587', '29']})

size_tpl = ("4 1/2 ", "7 ", "9 5/8 ", "13 3/8 ", "18 ", "30 ", "32 ")

# We break it into three parts from the end.
df['S,A,Q'] = df['SizeAgeQuantity'].str.rsplit(n=2)

# Incomplete lists are padded to three elements using the empty string.
mask_1 = (df['S,A,Q'].str.len() == 2) & (df['SizeAgeQuantity'].str.startswith(size_tpl))
if mask_1.any(): df['S,A,Q'].loc[mask_1].values[0].append("")

mask_2 = (df['S,A,Q'].str.len() == 2) & ~(df['SizeAgeQuantity'].str.startswith(size_tpl))
if mask_2.any(): df['S,A,Q'].loc[mask_2].values[0].insert(0, "")

mask_3 = (df['S,A,Q'].str.len() == 1) & (df['SizeAgeQuantity'].str.startswith(size_tpl))
if mask_3.any(): df['S,A,Q'].loc[mask_3].values[0].extend(("", ""))

mask_4 = (df['S,A,Q'].str.len() == 1) & ~(df['SizeAgeQuantity'].str.startswith(size_tpl))
if mask_4.any():
    df['S,A,Q'].loc[mask_4].values[0].insert(0, "")
    df['S,A,Q'].loc[mask_4].values[0].append("")

# We split the prepared column into the necessary columns and delete the ones that are no longer needed.
df[['Size', 'Age', 'Quantity']] = pd.DataFrame(df['S,A,Q'].values.tolist())
df.drop(columns=['SizeAgeQuantity', 'S,A,Q'], inplace=True)

print(df)

        Date  Item    Size   Age Quantity
0  2-18-2019  NY01  13 3/8     5      846
1  2-18-2019  Ld01   4 1/2   557       85
2  2-19-2019  Du02   9 5/8    47     4464
3  2-19-2019  Du01      30    58         
4  2-20-2019  Ps55      32   304      304
5  2-21-2019   L55       7  6588      685
6  2-21-2019  Du85          4118      587
7  2-22-2019   L85            29