1. Combinatorial insights into distributions of wealth, size, and abundance Ken Locey

5. Rank-abundance curve (RAC) Rank in abundance Abundance Frequency distribution Species abundance distribution (SAD) Abundance class frequency

6. Ranked curve (RC) Rank in abundance, wealth, or size Abundance/wealth/size Frequency distribution Distributions of wealth, size, abundance Abundance, wealth, or size class frequency

7. Wheat Production (tons)

8. Poverty in Rural America, 2008 Percent in Poverty 54 – 25.125 – 20.120 – 14.114 – 12.112 – 10.110 – 3.1

10. Distributions used to predict variation in wealth, size, & abundance 1.Pareto (80-20 rule) 2.Log-normal 3.Log-series 4.Geometric series 5.Dirichlet 6.Negative binomial 7.Zipf 8.Zipf-Mandelbrot

11. Rank-abundance curve (RAC) Rank in abundance Abundance Frequency distribution Predicting, modeling, & explaining the Species abundance distribution (SAD) Abundance class frequency

12. Rank in abundance Abundance 10 4 10 3 10 2 10 1 10 0 Observed Resource partitioning Demographic stochasticity Predicting, modeling, & explaining the Species abundance distribution (SAD)

13. Rank in abundance Abundance 10 4 10 3 10 2 10 1 10 0 N = 1,700 S = 17 Predicting, modeling, & explaining the Species abundance distribution (SAD)

14. How many forms of the SAD for a given N and S? Rank in abundance Abundance 10 4 10 3 10 2 10 1 10 0

15. Integer Partitioning Integer partition: A positive integer expressed as the sum of unordered positive integers e.g. 6 = 3+2+1 = 1+2+3 = 2+1+3 Written in non-increasing (lexical) order e.g. 3+2+1

16. Rank-abundance curves are integer partitions Rank-abundance curve N = total abundance S = species richness S unlabeled abundances that sum to N Integer partition N = positive integer S = number of parts S unordered +integers that sum to N =

17. Combinatorial Explosion NSShapes of the SAD 100010> 886 trillion 1000100> 302 trillion trillion

18. Random integer partitions Goal: Random partitions for N = 5, S = 3: 5 4+1 3+2 3+1+1 2+2+1 2+1+1+1 1+1+1+1+1 Nijenhuis and Wilf (1978) Combinatorial Algorithms for Computer and Calculators. Academic Press, New York.

19. SAD feasible sets are dominated by hollow curves Frequency log 2 (abundance)

20. The SAD feasible set ln(abundance ) Rank in abundance N=1000, S=40

21. Can we explain variation in abundance based on how N and S constrain observable variation? Question

22. Datasetcommunities Christmas Bird Count129 North American Breeding Bird Survey 1586 Gentry’s Forest Transect182 Forest Inventory & Analysis7359 Mammal Community Database42 Indoor Fungal Communities124 Terrestrial metagenomes 92 Aquatic metagenomes 48 TOTAL9562

23. The center of the feasible set ln(abundance) Rank in abundance N=1000, S=40

24. Observed abundance 10 0 10 1 10 2 Abundance at the center of the feasible set 10 2 10 1 10 0 R 2 per site R 2 = 1.0

25. Observed abundance R 2 = 0.93 Breeding Bird Survey (1,583 sites) 10 0 10 1 10 2 R 2 per site Abundance at the center of the feasible set 10 2 10 1 10 0

26. Abundance at center of the feasible set Observed abundance

27. Abundance at center of the feasible set

29. Public code and data repository https://github.com/weecology/feasiblesets

30. Center of the feasible set Observed home runs 0.930.88 0.91 0.940.93 http://mlb.mlb.com

31. Combinatorics is one only way to examine feasible sets Other (more common) ways: Mathematical optimization Linear programming

32. Datasettotal sitesanalyzable sites Christmas Bird Count1992129 (6.5%) North American Breeding Bird Survey 27691586 (57%) Gentry’s Forest Transect222182 (82%) Forest Inventory & Analysis103567359 (71%) Mammal Community Database10342 (41%) Indoor Fungal Communities128124 (97%) Terrestrial metagenomes 12892 (72%) Aquatic metagenomes 25248 (19%) TOTAL159509562 (60%)

33. Efficient algorithms for generating random integer partition with restricted numbers of parts

34. Random integer partitions Goal: Random partitions for N = 5, S = 3: 5 4+1 3+2 3+1+1 2+2+1 2+1+1+1 1+1+1+1+1 Nijenhuis and Wilf (1978) Combinatorial Algorithms for Computer and Calculators. Academic Press, New York.

35. Combinatorial Explosion NSSAD shapes 100010> 886 trillion 10001,...,1000> 2.4x10 31 Probability of generating a random partition of 1000 having 10 parts: < 10 -17

36. Task: Generate random partitions of N=9 having S=4 parts

37. 4+3+24+3+2

38. 4+3+24+3+2

39. 4+3+24+3+2

40. 4+3+24+3+2

41. 3+3+2+13+3+2+1 4+3+24+3+2

42. 4+3+24+3+2

43. 3+2=5

44. 4+3+2=9

45. 3+3+2+13+3+2+1

46. 1.Generate a random partition of N - S with S or less as the largest 2.Append S to the front 3.Conjugate the partition 4.Let cool & serve with garnish A recipe for random partitions of N with S parts

47. 5 4+1 3+2 3+1+1 2+2+1 2+1+1+1 1+1+1+1+1 Generate a random partition of N-S with S or less as the largest part Divide & Conquer Multiplicity Top down Bottom up

48. Un(bias) Skewness of partitions in a random sample Density

49. Speed Number of parts (S) Sage/algorithm N = 50N = 100 N = 150N = 200

50. Old Apples: probability of generating a partition for N = 1000 & S = 10: < 10 -17 New Oranges: Seconds to generate a partition for N = 1000 & S = 10: 0.07

51. Integer partitions S positive integers that sum to N without respect to order What if a distribution has zeros? subplots with 0 individuals people with 0 income publications with 0 citations

52. Abundance class frequency 012345 Intraspecific spatial abundance distribution (SSAD) N = abundance of a species S = number of subplots

53. Intraspecific spatial abundance distribution (SSAD)

54. Public code repository https://github.com/klocey/partitions PeerJ Preprint https://peerj.com/preprints/78/ Locey KJ, McGlinn DJ. (2013) Efficient algorithms for sampling feasible sets of macroecological patterns. PeerJ PrePrints 1:e78v1

55. Future Directions in Combinatorial Feasible Sets

56. Future Directions: metrics of Evenness, diversity, & inequality frequency

57. Future Directions: metrics of Evenness, diversity, & inequality frequency

58. Future Directions: metrics of Evenness, diversity, & inequality

59. Percentile in feasible setGini’s coefficient of inequality Future Directions: metrics of Evenness, diversity, & inequality

60. integer composition: all ordered ways that S positive integers can sum to N Future Directions: New combinatorial feasible sets 6 = 3+2+1 = 1+2+3 = 3+1+2

61. Future Directions: New combinatorial feasible sets Rank log abundance

62. Future Directions: New combinatorial feasible sets Rank log abundance

63. Future Directions: New combinatorial feasible sets Rank

64. Pragmatic: explanations & predictions using few inputs Mathematical: combinatorics can be used to characterize and understand observable variation in nature System specific: patterns attributed to specific processes are constrained by general variables. What drives the values of the variables? Policy, management, & philosophy: Would you want to know if the most costly, likely, preferred outcome was 95% similar to 95% of all others? Why?

65. http://figshare.com/articles/Combinatorial_i nsight_into_distributions_of_wealth_size_a nd_abundance/866822