Date: 201807
Local time of day: 1201-1800
Location: ZDV.ARTCC, CO
Aircraft: Citation Latitude (C680A)
Operating under: Part 91
Flight plan: IFR
Flight phase: Descent
ATC / advisory: Center ZDV
Second aircraft: MD-90 Series (DC-9-90) Undifferentiated or Other Model
Reporter organization: Government
Reporter function: Enroute
Reporter qualification: Air Traffic Control Fully Certified
Human factors: Workload; Situational Awareness; Communication Breakdown
Second reporter function: Captain
Anomaly: ATC Issue All Types; Conflict Airborne Conflict; Deviation - Altitude Excursion From Assigned Altitude; Deviation / Discrepancy - Procedural Clearance; Deviation / Discrepancy - Procedural Published Material / Policy; Inflight Event / Encounter Weather / Turbulence
Detector: Automation Air Traffic Control; Person Flight Crew
When detected: In-flight
Result: Air Traffic Control Issued New Clearance; Flight Crew Rejected Takeoff; Flight Crew Requested ATC Assistance / Clarification
Contributing factors: ATC Equipment / Nav Facility / Buildings; Staffing; Procedure; Human Factors
Primary problem: Procedure

Narrative: I think Sector X was red - or close to it - during our noon push. I was also handling multiple deviations for weather. Our staffing is low; so we did not have a D side at the time to offer me and our supervisor was in recurrent training so we had a CIC on the desk; taking one more controller away from the floor. I felt busy - I descended Aircraft X from FL410 to FL390 to start stepping him down through all my traffic. I think I was busy enough that I don't remember the read back. I continued working on other things and saw conflict alert go off as the Aircraft X pilot asked me if he had traffic. I said yes - and told him I thought I had cleared him to FL390 - he said he answered with FL290. He was a few miles - maybe 3.5 miles lateral from Aircraft Y at FL380. I saw more traffic for him at FL360 so I reissued his clearance to descend to FL370. He repeated FL370.We need better staffing. We shouldn't be allowing controllers or supervisors to attend classes when we are 1 below the minimum safe number of controllers at the start of the shift. If I had a second pair of eyes or ears; maybe they would have caught it. I haven't listened to the FALCON; but I probably did miss his read back. I don't know.


Second narrative: Just north of Moab; Denver Center cleared [our flight] to descend to FL290. We read back FL290. We descended from FL410 and approx. FL378; I noticed traffic on TCAS around FL370. I stopped descent and told ATC we have traffic on TCAS. [There was] no TA/RA. Controller replied 'say altitude'. I replied 'descending thru FL378 for FL290'. Controller replied 'I thought I gave you FL390'. I said 'no you gave us FL290'. Controller replied 'descend and maintain FL370'. I listened to our onboard playback of ATC transmissions and Controller clearly said descend and maintain FL290. When this event occurred; we were within 6 miles and 700 ft of traffic. No RA occurred. Additionally; I stopped the descent when I realized we were descending on the traffic's altitude.


Synopsis: Denver Center Controller and corporate pilot reported a loss of separation due to communication error with pilot and ATC.