Explicit Polynomial expressions, in, n, for The Expectation, Variance, and first , 6, moments (about the mean) of the Size of an (2n+1,2n+3)-core partition with distinct parts By Shalosh B. Ekhad Armin Straub (arXiv: math.CO 1601.07161v) conjectured that the number of (2n+1,2n+3)-core partitions into distinct parts is given by the simple formula 4^n. This was first proved by Sherry H.F. Yan, Guizhi Qin, Zemin Jin, and Robin D.P. Zhou (arXiv: math.CO 1604.03729v1), who gave an ingenious but complicated proof. But what about the average size, variance, and the higher moments? In this article we will state explicit polynomial expressions for the average, variance, all the way to the 6-th moment, and will also give explicit numerical values for the limits of the scaled moments. Theorem 1: The average size of an (2n+1,2n+3)-core partition into distinct parts is 19 27 2 3 -- n + -- n + 5/16 n 32 32 and in Maple notation 19/32*n+27/32*n^2+5/16*n^3 Theorem 2: The variance of size of an (2n+1,2n+3)-core partition into distinct parts is 253 391 2 2101 3 485 4 4687 5 467 6 ---- n + --- n + ---- n + --- n + ----- n + ---- n 2560 960 3072 768 15360 7680 and in Maple notation 253/2560*n+391/960*n^2+2101/3072*n^3+485/768*n^4+4687/15360*n^5+467/7680*n^6 Hence the limit of the coefficient of variation, as n goes to infinity is 1/2 14010 -------- 150 and in floating-point, 0.7890923054 Since this is NOT zero, there is NO Concentration about the mean Theorem , 3, : The , 3, -th moment (about the mean) of the size of an (2n+1,2n+3)-core partition into distinct parts is 27037 1085653 2 1189201 3 43883 4 169135 5 1229663 6 - ------ n - ------- n - ------- n + ------ n + ------ n + ------- n 573440 6881280 6881280 786432 393216 1966080 1596893 7 989005 8 396793 9 + ------- n + ------- n + -------- n 3440640 5505024 13762560 and in Maple notation -27037/573440*n-1085653/6881280*n^2-1189201/6881280*n^3+43883/786432*n^4+169135 /393216*n^5+1229663/1966080*n^6+1596893/3440640*n^7+989005/5505024*n^8+396793/ 13762560*n^9 Hence the limit of the scaled , 3, -th moment (alpha coefficient) is 1/2 1/2 396793 467 7680 --------------------- 390815488 and in floating-point 1.922787480 This is called the limiting skewness Since it is not zero, note that the size is NOT asymptotically normal Theorem , 4, : The , 4, -th moment (about the mean) of the size of an (2n+1,2n+3)-core partition into distinct parts is 5743943 212460209 2 74298451 3 1985069 4 89856527 5 -------- n + ---------- n + ---------- n + -------- n + --------- n 75694080 1513881600 1238630400 12386304 141557760 2063105059 6 82066319 7 330827333 8 1592156089 9 + ---------- n + -------- n + --------- n + ---------- n 1651507200 45875200 165150720 990904320 4128011389 10 963591731 11 2421823 12 + ---------- n + ---------- n + -------- n 4954521600 3892838400 75694080 and in Maple notation 5743943/75694080*n+212460209/1513881600*n^2+74298451/1238630400*n^3+1985069/ 12386304*n^4+89856527/141557760*n^5+2063105059/1651507200*n^6+82066319/45875200 *n^7+330827333/165150720*n^8+1592156089/990904320*n^9+4128011389/4954521600*n^ 10+963591731/3892838400*n^11+2421823/75694080*n^12 Hence the limit of the scaled , 4, -th moment (alpha coefficient) is 145309380 --------- 16792853 and in floating-point 8.653049008 This is called the limiting kurtosis Since it is not 3, note that this is another reason why size is NOT asymptotically normal Theorem , 5, : The , 5, -th moment (about the mean) of the size of an (2n+1,2n+3)-core partition into distinct parts is 941102495 166709531203 2 1252879634117 3 258229299487 4 - ---------- n - ------------ n + ------------- n + ------------ n 1574436864 188932423680 2267189084160 209278992384 37406122921 5 100406451571 6 370758594737 7 73572561695 8 - ----------- n - ------------ n - ------------ n + ----------- n 65399685120 54358179840 761014517760 50734301184 80590902421 9 1967122658101 10 23724020823893 11 + ----------- n + ------------- n + -------------- n 36238786560 761014517760 8371159695360 543168228587 12 123571765365781 13 1275815000501 14 + ------------ n + --------------- n + ------------- n 239175991296 108825076039680 4030558371840 685932873011 15 + -------------- n 18137512673280 and in Maple notation -941102495/1574436864*n-166709531203/188932423680*n^2+1252879634117/ 2267189084160*n^3+258229299487/209278992384*n^4-37406122921/65399685120*n^5-\ 100406451571/54358179840*n^6-370758594737/761014517760*n^7+73572561695/ 50734301184*n^8+80590902421/36238786560*n^9+1967122658101/761014517760*n^10+ 23724020823893/8371159695360*n^11+543168228587/239175991296*n^12+ 123571765365781/108825076039680*n^13+1275815000501/4030558371840*n^14+ 685932873011/18137512673280*n^15 Hence the limit of the scaled , 5, -th moment (alpha coefficient) is 1/2 1/2 3429664365055 467 7680 ---------------------------- 156594294624768 and in floating-point 41.47770671 Theorem , 6, : The , 6, -th moment (about the mean) of the size of an (2n+1,2n+3)-core partition into distinct parts is 51993347431169 366779094930757 2 1367298243590759 3 -------------- n + --------------- n - ---------------- n 4282468270080 18447555624960 181375126732800 108070084446305647 4 11459603281568807 5 66841782804596623 6 - ------------------ n - ----------------- n + ----------------- n 3385669032345600 870600608317440 4062802838814720 5201723829684161 7 56834727032487689 8 952734300032257 9 + ---------------- n - ----------------- n - --------------- n 334846387814400 12500931811737600 97409858273280 336468594383711 10 16520393854479871 11 21883859279244619 12 + --------------- n + ----------------- n + ----------------- n 113644834652160 1339385551257600 2083488635289600 6248335270247533 13 732571946795803 14 61587508650342319 15 + ---------------- n + --------------- n + ----------------- n 994972123791360 156261647646720 17412012166348800 286051737202779731 16 12657012510344537 17 15302567641785799 18 + ------------------ n + ----------------- n + ------------------ n 162512113552588800 26311485051371520 276270593039400960 and in Maple notation 51993347431169/4282468270080*n+366779094930757/18447555624960*n^2-\ 1367298243590759/181375126732800*n^3-108070084446305647/3385669032345600*n^4-\ 11459603281568807/870600608317440*n^5+66841782804596623/4062802838814720*n^6+ 5201723829684161/334846387814400*n^7-56834727032487689/12500931811737600*n^8-\ 952734300032257/97409858273280*n^9+336468594383711/113644834652160*n^10+ 16520393854479871/1339385551257600*n^11+21883859279244619/2083488635289600*n^12 +6248335270247533/994972123791360*n^13+732571946795803/156261647646720*n^14+ 61587508650342319/17412012166348800*n^15+286051737202779731/162512113552588800* n^16+12657012510344537/26311485051371520*n^17+15302567641785799/ 276270593039400960*n^18 Hence the limit of the scaled , 6, -th moment (alpha coefficient) is 382564191044644975 ------------------ 1552893421695616 and in floating-point 246.3557291 To summarize, here is the list of moments, starting from the average 19 27 2 3 [-- n + -- n + 5/16 n , 32 32 253 391 2 2101 3 485 4 4687 5 467 6 27037 ---- n + --- n + ---- n + --- n + ----- n + ---- n , - ------ n 2560 960 3072 768 15360 7680 573440 1085653 2 1189201 3 43883 4 169135 5 1229663 6 - ------- n - ------- n + ------ n + ------ n + ------- n 6881280 6881280 786432 393216 1966080 1596893 7 989005 8 396793 9 5743943 212460209 2 + ------- n + ------- n + -------- n , -------- n + ---------- n 3440640 5505024 13762560 75694080 1513881600 74298451 3 1985069 4 89856527 5 2063105059 6 82066319 7 + ---------- n + -------- n + --------- n + ---------- n + -------- n 1238630400 12386304 141557760 1651507200 45875200 330827333 8 1592156089 9 4128011389 10 963591731 11 + --------- n + ---------- n + ---------- n + ---------- n 165150720 990904320 4954521600 3892838400 2421823 12 941102495 166709531203 2 1252879634117 3 + -------- n , - ---------- n - ------------ n + ------------- n 75694080 1574436864 188932423680 2267189084160 258229299487 4 37406122921 5 100406451571 6 370758594737 7 + ------------ n - ----------- n - ------------ n - ------------ n 209278992384 65399685120 54358179840 761014517760 73572561695 8 80590902421 9 1967122658101 10 23724020823893 11 + ----------- n + ----------- n + ------------- n + -------------- n 50734301184 36238786560 761014517760 8371159695360 543168228587 12 123571765365781 13 1275815000501 14 + ------------ n + --------------- n + ------------- n 239175991296 108825076039680 4030558371840 685932873011 15 51993347431169 366779094930757 2 + -------------- n , -------------- n + --------------- n 18137512673280 4282468270080 18447555624960 1367298243590759 3 108070084446305647 4 11459603281568807 5 - ---------------- n - ------------------ n - ----------------- n 181375126732800 3385669032345600 870600608317440 66841782804596623 6 5201723829684161 7 56834727032487689 8 + ----------------- n + ---------------- n - ----------------- n 4062802838814720 334846387814400 12500931811737600 952734300032257 9 336468594383711 10 16520393854479871 11 - --------------- n + --------------- n + ----------------- n 97409858273280 113644834652160 1339385551257600 21883859279244619 12 6248335270247533 13 732571946795803 14 + ----------------- n + ---------------- n + --------------- n 2083488635289600 994972123791360 156261647646720 61587508650342319 15 286051737202779731 16 12657012510344537 17 + ----------------- n + ------------------ n + ----------------- n 17412012166348800 162512113552588800 26311485051371520 15302567641785799 18 + ------------------ n ] 276270593039400960 and in Maple notation [19/32*n+27/32*n^2+5/16*n^3, 253/2560*n+391/960*n^2+2101/3072*n^3+485/768*n^4+ 4687/15360*n^5+467/7680*n^6, -27037/573440*n-1085653/6881280*n^2-1189201/ 6881280*n^3+43883/786432*n^4+169135/393216*n^5+1229663/1966080*n^6+1596893/ 3440640*n^7+989005/5505024*n^8+396793/13762560*n^9, 5743943/75694080*n+ 212460209/1513881600*n^2+74298451/1238630400*n^3+1985069/12386304*n^4+89856527/ 141557760*n^5+2063105059/1651507200*n^6+82066319/45875200*n^7+330827333/ 165150720*n^8+1592156089/990904320*n^9+4128011389/4954521600*n^10+963591731/ 3892838400*n^11+2421823/75694080*n^12, -941102495/1574436864*n-166709531203/ 188932423680*n^2+1252879634117/2267189084160*n^3+258229299487/209278992384*n^4-\ 37406122921/65399685120*n^5-100406451571/54358179840*n^6-370758594737/ 761014517760*n^7+73572561695/50734301184*n^8+80590902421/36238786560*n^9+ 1967122658101/761014517760*n^10+23724020823893/8371159695360*n^11+543168228587/ 239175991296*n^12+123571765365781/108825076039680*n^13+1275815000501/ 4030558371840*n^14+685932873011/18137512673280*n^15, 51993347431169/ 4282468270080*n+366779094930757/18447555624960*n^2-1367298243590759/ 181375126732800*n^3-108070084446305647/3385669032345600*n^4-11459603281568807/ 870600608317440*n^5+66841782804596623/4062802838814720*n^6+5201723829684161/ 334846387814400*n^7-56834727032487689/12500931811737600*n^8-952734300032257/ 97409858273280*n^9+336468594383711/113644834652160*n^10+16520393854479871/ 1339385551257600*n^11+21883859279244619/2083488635289600*n^12+6248335270247533/ 994972123791360*n^13+732571946795803/156261647646720*n^14+61587508650342319/ 17412012166348800*n^15+286051737202779731/162512113552588800*n^16+ 12657012510344537/26311485051371520*n^17+15302567641785799/276270593039400960*n ^18] and this is the list of limiting scled moments , starting with the coefficie\ nt of variation 1/2 1/2 1/2 1/2 1/2 14010 396793 467 7680 145309380 3429664365055 467 7680 [--------, ---------------------, ---------, ----------------------------, 150 390815488 16792853 156594294624768 382564191044644975 ------------------] 1552893421695616 and in Maple format [1/150*14010^(1/2), 396793/390815488*467^(1/2)*7680^(1/2), 145309380/16792853, 3429664365055/156594294624768*467^(1/2)*7680^(1/2), 382564191044644975/ 1552893421695616] and in floating-point [.7890923054, 1.922787480, 8.653049008, 41.47770671, 246.3557291] This ends this article, that took, 15.993, seconds to generate.