Søren RiisQueen Mary University of London

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Explore the data behind the results, reuse research software, and inspect formal proofs.

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The Condorcet atlas

A census of 171,870,480 representatives on seven alternatives. See the distribution, compare its extremes, and inspect the rankings that keep majority choice consistent.

All 3,353 examples of size ≤16 and all 4,679 examples of size ≥88 are available here.

Explore the atlas →Read the paper
Logarithmic count scale. The peak is at size 56. Sizes at most 16 and at least 88 have examples included here; the complete numeric summary follows.110010k1m100mSize 4: 46 representativesSize 8: 44 representativesSize 9: 24 representativesSize 10: 270 representativesSize 11: 188 representativesSize 12: 147 representativesSize 13: 176 representativesSize 14: 284 representativesSize 15: 548 representativesSize 16: 1,626 representativesSize 17: 2,178 representativesSize 18: 4,435 representativesSize 19: 9,994 representativesSize 20: 11,864 representativesSize 21: 7,040 representativesSize 22: 12,685 representativesSize 23: 18,570 representativesSize 24: 25,954 representativesSize 25: 44,660 representativesSize 26: 81,579 representativesSize 27: 94,158 representativesSize 28: 132,114 representativesSize 29: 159,868 representativesSize 30: 194,674 representativesSize 31: 247,692 representativesSize 32: 404,009 representativesSize 33: 356,618 representativesSize 34: 480,195 representativesSize 35: 461,900 representativesSize 36: 609,624 representativesSize 37: 678,422 representativesSize 38: 928,441 representativesSize 39: 930,304 representativesSize 40: 1,244,522 representativesSize 41: 1,273,738 representativesSize 42: 1,739,772 representativesSize 43: 1,849,074 representativesSize 44: 2,701,280 representativesSize 45: 2,644,266 representativesSize 46: 3,491,780 representativesSize 47: 3,686,966 representativesSize 48: 4,911,214 representativesSize 49: 4,790,868 representativesSize 50: 6,426,642 representativesSize 51: 6,253,444 representativesSize 52: 8,174,653 representativesSize 53: 7,497,364 representativesSize 54: 9,180,598 representativesSize 55: 8,270,608 representativesSize 56: 11,160,909 representativesSize 57: 8,540,924 representativesSize 58: 10,269,782 representativesSize 59: 7,723,932 representativesSize 60: 9,606,176 representativesSize 61: 6,518,148 representativesSize 62: 7,839,946 representativesSize 63: 5,191,166 representativesSize 64: 7,728,718 representativesSize 65: 3,436,076 representativesSize 66: 3,750,621 representativesSize 67: 2,034,070 representativesSize 68: 2,440,206 representativesSize 69: 1,152,526 representativesSize 70: 1,351,871 representativesSize 71: 671,796 representativesSize 72: 808,375 representativesSize 73: 357,970 representativesSize 74: 405,334 representativesSize 75: 186,106 representativesSize 76: 244,369 representativesSize 77: 101,268 representativesSize 78: 116,958 representativesSize 79: 48,120 representativesSize 80: 56,464 representativesSize 81: 23,402 representativesSize 82: 25,154 representativesSize 83: 11,344 representativesSize 84: 14,503 representativesSize 85: 6,108 representativesSize 86: 4,273 representativesSize 87: 2,066 representativesSize 88: 2,038 representativesSize 89: 1,248 representativesSize 90: 647 representativesSize 91: 214 representativesSize 92: 274 representativesSize 93: 106 representativesSize 94: 76 representativesSize 95: 18 representativesSize 96: 36 representativesSize 97: 16 representativesSize 98: 4 representativesSize 100: 2 representatives120406080100RepresentativesDomain size · rankings

Number of archived representatives by domain size · logarithmic scale

The full size summary and the selected collections are hosted on this website. The explorer loads a collection only when requested. The authors’ archive provides the complete downloads.

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From results to reusable tools

Research software & formal proofs

CDL

A library for Condorcet domains and restricted permutation sets, with C++ code and Python bindings.

Bei Zhou, Klas Markström & Søren Riis
SoftwareX 28 (2024), 101951

Code & documentation ↗Software paper

The paper, unfolded

A theory paper with paragraph-level explanations, exact Lean statements, and proof steps that can be inspected alongside the argument.

One Unit Separates Polynomial Time from Undecidability in Term Coding · Søren Riis, revised preprint

Explore the paper ↗Source & formal proofsHosted PDF

A view of the wider field

Combinatorics submissions over time

Klas Markström’s chart compares weekly arXiv submissions in combinatorics across years, with annotations for selected AI releases and mathematical milestones.

Klas Markström’s chart of weekly arXiv math.CO submissions for 2022–2026, using a four-week running average. The 2026 line rises above earlier years in the displayed period; selected AI events are annotated.
Open the full-size chart. Chart by Klas Markström. Local snapshot: 13 September 2026. View the source chart ↗. The lines measure submission volume. The event annotations provide a timeline; the chart alone does not establish that AI caused a change in submissions, or measure the quality of the papers.