Enroute Controller described a busy/complex traffic period during which supervisors failed to provide assistance allegedly because of their reliance on the TSD in lieu of position observations.

Narrative

I relieved the Controller; working low to moderate traffic. Within the next five to fifteen minutes I was working fifteen to eighteen aircraft; high volume and complexity. Several aircraft were descending and climbing per LOA. I had a negative RVSM above the airspace and several ZZZ landers needing descent; with a ZZZ1 departure climbing from FL310 to FL410 southwest bound. The complexity was very high; yet I was required to continue working a combined position. The FLM did not leave his desk to provide or ask if assistance was needed. I was too busy to ask for assistance. Several aircraft; three to six; were handed-off; but not shipped to the adjacent sectors prior to five to ten miles into their airspace. An adjacent Center called on two separate aircraft stating they had not been transferred. I was unable to maintain sector integrity because of complexity with several other aircraft in conflict and on vectors. An aircraft was left too high and asked for lower. I am a long time veteran; traffic is down; fatigue mitigation implementation is a major distraction; causing FLMs to lose track of priorities; but we have the staffing to avoid these situations. The TSD display is overly used by FLMs to manage staffing. It causes situations to happen without their awareness. Sitting at the desk is a major cause of a lack of situational awareness. Unless staffing is at a minimum; which can happen; staff sectors based on real observations and not the TSD or OM directives. Make FLMs more accountable for staffing decisions.