The generating function of the number of reduced decompostions of , [1], is , 1 and in Maple notation 1 The expectation is, 0, and the variance is, 0 The generating function of the number of reduced decompostions of , [2, 1], is , 1 and in Maple notation 1 The expectation is, 0, and the variance is, 0 The generating function of the number of reduced decompostions of , [3, 2, 1], is , 2 x and in Maple notation 2*x The expectation is, 1, and the variance is, 0 The generating function of the number of reduced decompostions of , 2 [4, 3, 2, 1], is , 4 x + 8 x + 4 and in Maple notation 4*x^2+8*x+4 The expectation is, 1, and the variance is, 1/2 The generating function of the number of reduced decompostions of , 3 2 [5, 4, 3, 2, 1], is , 32 x + 192 x + 288 x + 256 and in Maple notation 32*x^3+192*x^2+288*x+256 The expectation is, 1, and the variance is, 3/4 The generating function of the number of reduced decompostions of , [6, 5, 4, 3, 2, 1], is , 6 5 4 3 2 4 x + 56 x + 2152 x + 16864 x + 59780 x + 113800 x + 100208 and in Maple notation 4*x^6+56*x^5+2152*x^4+16864*x^3+59780*x^2+113800*x+100208 11 The expectation is, 1, and the variance is, -- 13 The generating function of the number of reduced decompostions of , 7 6 5 4 [7, 6, 5, 4, 3, 2, 1], is , 1456 x + 67064 x + 1236400 x + 11741236 x 3 2 + 64411616 x + 215354704 x + 423238880 x + 384691300 and in Maple notation 1456*x^7+67064*x^6+1236400*x^5+11741236*x^4+64411616*x^3+215354704*x^2+ 423238880*x+384691300 17 The expectation is, 1, and the variance is, -- 19 The generating function of the number of reduced decompostions of , 10 9 8 [8, 7, 6, 5, 4, 3, 2, 1], is , 1760 x + 262160 x + 16221264 x 7 6 5 4 + 459797952 x + 7673875968 x + 82084933968 x + 589039596960 x 3 2 + 2874227383136 x + 9331832099808 x + 18506472293632 x + 17216989222352 and in Maple notation 1760*x^10+262160*x^9+16221264*x^8+459797952*x^7+7673875968*x^6+82084933968*x^5+ 589039596960*x^4+2874227383136*x^3+9331832099808*x^2+18506472293632*x+ 17216989222352 12 The expectation is, 1, and the variance is, -- 13 The generating function of the number of reduced decompostions of , 13 12 11 [9, 8, 7, 6, 5, 4, 3, 2, 1], is , 10960 x + 5034544 x + 698402288 x 10 9 8 + 42817670128 x + 1525315547632 x + 35652331972512 x 7 6 5 + 581895590060208 x + 6862185937112976 x + 59435324031326192 x 4 3 2 + 378559369042744528 x + 1743764520961564288 x + 5555432536259193248 x + 11059248497359308336 x + 10454445445908540480 and in Maple notation 10960*x^13+5034544*x^12+698402288*x^11+42817670128*x^10+1525315547632*x^9+ 35652331972512*x^8+581895590060208*x^7+6862185937112976*x^6+59435324031326192*x ^5+378559369042744528*x^4+1743764520961564288*x^3+5555432536259193248*x^2+ 11059248497359308336*x+10454445445908540480 16 The expectation is, 1, and the variance is, -- 17 The generating function of the number of reduced decompostions of , 16 15 [10, 9, 8, 7, 6, 5, 4, 3, 2, 1], is , 2902744 x + 1414517032 x 14 13 12 + 265564910080 x + 25719037598280 x + 1513067459993136 x 11 10 + 60324617640192008 x + 1735043173906471128 x 9 8 + 37378865919043503688 x + 617236288685567710064 x 7 6 + 7914881286496535714224 x + 79192602595069201432248 x 5 4 + 616121420454158913885904 x + 3677078633785350968331576 x 3 2 + 16365312063606437486059200 x + 51475549145956695216389904 x + 102663472426214544974013184 x + 98149983077575319554367920 and in Maple notation 2902744*x^16+1414517032*x^15+265564910080*x^14+25719037598280*x^13+ 1513067459993136*x^12+60324617640192008*x^11+1735043173906471128*x^10+ 37378865919043503688*x^9+617236288685567710064*x^8+7914881286496535714224*x^7+ 79192602595069201432248*x^6+616121420454158913885904*x^5+ 3677078633785350968331576*x^4+16365312063606437486059200*x^3+ 51475549145956695216389904*x^2+102663472426214544974013184*x+ 98149983077575319554367920 41 The expectation is, 1, and the variance is, -- 43 To sum up, the first , 10, generating functions are [1, 1, 2*x, 4*x^2+8*x+4, 32*x^3+192*x^2+288*x+256, 4*x^6+56*x^5+2152*x^4+16864* x^3+59780*x^2+113800*x+100208, 1456*x^7+67064*x^6+1236400*x^5+11741236*x^4+ 64411616*x^3+215354704*x^2+423238880*x+384691300, 1760*x^10+262160*x^9+16221264 *x^8+459797952*x^7+7673875968*x^6+82084933968*x^5+589039596960*x^4+ 2874227383136*x^3+9331832099808*x^2+18506472293632*x+17216989222352, 10960*x^13 +5034544*x^12+698402288*x^11+42817670128*x^10+1525315547632*x^9+35652331972512* x^8+581895590060208*x^7+6862185937112976*x^6+59435324031326192*x^5+ 378559369042744528*x^4+1743764520961564288*x^3+5555432536259193248*x^2+ 11059248497359308336*x+10454445445908540480, 2902744*x^16+1414517032*x^15+ 265564910080*x^14+25719037598280*x^13+1513067459993136*x^12+60324617640192008*x ^11+1735043173906471128*x^10+37378865919043503688*x^9+617236288685567710064*x^8 +7914881286496535714224*x^7+79192602595069201432248*x^6+ 616121420454158913885904*x^5+3677078633785350968331576*x^4+ 16365312063606437486059200*x^3+51475549145956695216389904*x^2+ 102663472426214544974013184*x+98149983077575319554367920] ------------------------------------ This took, 20516.266, seconds.