This is a bit of a thematic reach, but in honor of America's Independence Day, here's a simple cotillion mixer figure featuring American flags. It is taken from H. Layton Walker's Twentieth Century Cotillion Figures, (Buffalo, NY: Two Step Publishing Company, 1912), which compiled figures from a variety of different sources. The diagram at left is from Walker, in which it is arranged for fifty-seven couples, but the figure can easily be performed by a smaller (or larger) number.
The essence of the figure is to divide the dancing couples into groups of increasing size, with each group in turn (smallest to largest) first two-stepping and then, at a signal, forming a circle as the next group begins to dance. Each new circle forms around the previous one(s) and circles in the opposite direction. Once each group is circling, a signal starts the dancers doing a grand right and left in their own circle -- this is the mixer element -- and then, at the last signal, taking the person they are now facing as their partner for a final two-step.
So where do the flags come in? As part of the division of the couples into groups, each group receives a different flag. As arranged for fifty-seven couples, the groups are:
3 couples - American flags, circle right
6 couples - other national flags, circle left
10 couples - red flags, circle right
16 couples - white flags, circle left
22 couples - blue flags, circle right
With a smaller number of couples, just adjust the flag selection and/or quantities and number of circles accordingly. One could even use all American flags, varying them by attaching different colored ribbons or tassels.
No specific tune is given, but Sousa's "The Stars and Stripes Forever" (1896) would be appropriate. Here's a video set to a rare live recording of Sousa first speaking and then conducting the piece in 1929:
A final note: I'm quite aware that the UK ceremony performed on the monarch's official birthday is "Trooping the Colour" (note spelling difference: trooping vs. trouping). Either Walker or his source got it wrong.
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Transcription of the original description
TROUPING THE COLORS
Arranged for any number of couples. Arranged for 57 couples.
First, march past four tables in 3, 5, 10, 16, and 22 couples.
First three couples receive U. S. A. flags.
Next six couples receive flags of all nations.
Next 10 couples receive plain red flags.
Next 16 couples receive plain white flags.
Next 22 couples receive plain blue flags.
First signal, three couples dance (two-step.)
Second signal, three couples circle (to right) in centre of hall, and six couples dance.
Third signal, six couples circle (to left) around three couples and ten couples dance.
Fourth signal, 10 couples circle (to right) around six couples and 16 couples dance.
Fifth signal, 16 couples circle (to left) around 10 couples and 22 couples dance.
Sixth signal, blue flags around all, circle to right at diagram.
Grand right and left in your own circle and dance at signal.
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