Counting Langford pairings for 31 and 32 pairs

Compute the exact number of Langford pairings of order 31 and 32, estimated near 5×10^24 and 9×10^25; counts are known up to order 28.

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File it in the catalog (shoal-catalog 0521c19bbd38)

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	1	+{
	2	+ "domain": "computer-science",
	3	+ "topics": [
	4	+ "constraint-programming",
	5	+ "algorithms-complexity"
	6	+ ],
	7	+ "type": "algorithm",
	8	+ "source": {
	9	+ "key": "csplib",
	10	+ "name": "CSPLib",
	11	+ "url": "https://www.csplib.org/Problems/prob024/",
	12	+ "id": "prob024",
	13	+ "curator": "CSPLib editors",
	14	+ "poser": "Toby Walsh",
	15	+ "status": "open",
	16	+ "as_of": "2026-10-20",
	17	+ "catalog": "0521c19bbd38"
	18	+ }
	19	+}