If the 1-3 shoots that make up a division were taken from a mature plant in late winter/early spring and those shoots had not initiated flower buds the prior year, that division will not flower. In fact some species like S. leucophylla, if divided down to single growing tip pieces from a large mature clump will often sulk for a year or two, growing very little and very rarely if ever flower. Remember that a "small" division from a mature plant is not a mature plant itself.
Realize that flower bud initiation in Sarracenia usually occurs sometime in mid-late summer of the previous year to when you see the flower. It likely varies by species and might be in early fall for some. The overall good health/vigor of the plant and energy reserves built up during the late spring and summer are what contribute to the flower bud development. This is likely the primary reason many sun-requiring plants like Sarracenia fail to flower when they become overly shaded. Once formed, the flower buds remain dormant in the apices over the winter where it is likely they need some amount of vernalization (exposure to prolonged cool temps in the range of 32deg F to say 40deg F for example) to properly develop. Like tulips that need to be chilled to flower properly in the south. After passing through the dormant “cold” phase, a period of warmth is then needed to initiate final development of the floral structures and elongation of the stem and opening of the flower.
I suspect that the needed amount of heat doesn’t seem to be very great, as many of us in the PNW have been experiencing one of the cooler and wetting springs and summers in recent years. Last summer (when flower buds were initiating) was relatively cool and my plants in pots which sat outside in wading pools in full sun all summer and came though the week long freezes where the minimum temps reached 10deg F in November, 11deg F in December and 13deg F in February with no protection, flowered quite heavily this year though they were a good 4-8 weeks behind schedule.
It is not uncommon to obtain a “small” division that flowers that same year. The flower bud was initiated the previous year. This will put a bit of strain on a small division, which is why it is often recommend to remove such buds immediately upon detection. If the plant doesn’t produce enough leaves and receive enough sun, it often will not initiate a bud that summer, thus no flowering the following year. In some bulbs like Hippeastrum (Christmas Amaryllis), there are a very specific minimum number of leaves needed to properly develop a flower bud. I don’t know what that might be for Sarracenia but I can say that my red-tube S. flava, which only puts out 1-3 pitchers and 2-3 phyllodia per year, has never flowered in the six years I have grown it.
Bottom line is, for maximum flowering ensure the plants grow as many leaves as possible, are in full sun, never want for water, consume a reasonable amount of prey but avoid the sin of gluttony and above all be patient.
Always end an explanation with another question. Assume a healthy, vigorous multi-shoot Sarracenia that has flowered well in the past. Does any single shoot that bears a flower and then sets a good seed pod that goes to maturity, flower then next year? Does it "rest" a year? In many plants like apples and rhododendrons, if a fruit or seed pod forms at the end of shoot then no flower bud is initiated thus no bloom on that stem the next year.