Partitioning the Digit

Just brute force computer work: $\text{Length}[\text{Table}[\text{With}[\{l=\text{IntegerDigits}[n]\},\text{If}[l[[1]]=\text{Plus}\text{@@}\text{Drop}[l,1],n,\text{Nothing}]],\{n,1000,9999\}]] \Rightarrow 219$

{1001,1010,1100,2002,2011,2020,2101,2110,2200,3003,3012,3021,3030,3102,3111,3120,3201,3210,3300,4004,4013,4022,4031,4040,4103, 4112,4121,4130,4202,4211,4220,4301,4310,4400,5005,5014,5023,5032,5041,5050,5104,5113,5122,5131,5140,5203,5212,5221,5230,5302, 5311,5320,5401,5410,5500,6006,6015,6024,6033,6042,6051,6060,6105,6114,6123,6132,6141,6150,6204,6213,6222,6231,6240,6303,6312, 6321,6330,6402,6411,6420,6501,6510,6600,7007,7016,7025,7034,7043,7052,7061,7070,7106,7115,7124,7133,7142,7151,7160,7205,7214, 7223,7232,7241,7250,7304,7313,7322,7331,7340,7403,7412,7421,7430,7502,7511,7520,7601,7610,7700,8008,8017,8026,8035,8044,8053, 8062,8071,8080,8107,8116,8125,8134,8143,8152,8161,8170,8206,8215,8224,8233,8242,8251,8260,8305,8314,8323,8332,8341,8350,8404, 8413,8422,8431,8440,8503,8512,8521,8530,8602,8611,8620,8701,8710,8800,9009,9018,9027,9036,9045,9054,9063,9072,9081,9090,9108, 9117,9126,9135,9144,9153,9162,9171,9180,9207,9216,9225,9234,9243,9252,9261,9270,9306,9315,9324,9333,9342,9351,9360,9405,9414, \9423,9432,9441,9450,9504,9513,9522,9531,9540,9603,9612,9621,9630,9702,9711,9720,9801,9810,9900}